ORIGINAL_ARTICLE
Mathematical Chemistry Works of Dragos Cvetkovic
In addition to his countless contributions to spectral graph theory, some works of Dragos Cvetkovic are concerned with chemical problems. These are briefly outlined, with emphasis on his collaboration with the present author.
https://mir.kashanu.ac.ir/article_95507_bbf21ebf318b0d63b6a7ed64364739ab.pdf
2019-12-01
129
136
10.22052/mir.2019.204819.1168
Spectral graph theory
Mhemical graph theory
molecular graph
Huckel molecular orbital theory
Ivan
Gutman
gutman@kg.ac.rs
1
Faculty of Science, University of Kragujevac, Kragujevac, Serbia
LEAD_AUTHOR
[1] K. Balinska, D. Cvetkovic, M. Lepovic and S. Simic, There are exactly 150 connected integral graphs up to 10 vertices, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999) 95–105.
1
[2] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs – Theory and Applications, Third Edition, Johann Ambrosius Barth, Heidelberg, 1995.
2
[3] D. Cvetkovic, P. Rowlinson and S. Simic, An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts, 75, Cambridge Univ. Press, Cambridge, 2010.
3
[4] D. M. Cvetkovic and I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Mat. Vesnik 9 (1972) 141–150.
4
[5] D. Cvetkovic, I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. II, Croat. Chem. Acta 44 (1972) 365–374.
5
[6] D. Cvetkovic, I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. IX. On the stability of cata-condensed hydrocarbons, Theor. Chim. Acta 34 (1974) 129–136.
6
[7] D. Cvetkovic, I. Gutman and N. Trinajstic, Graphical studies on the relations between the structure and reactivity of conjugated systems: The role of nonbonding molecular orbitals, J. Mol. Struct. 28 (1975) 289–303.
7
[8] D. Cvetkovic and I. Gutman, Kekulé structures and topology. II. Catacondensed systems, Croat. Chem. Acta 46 (1974) 15–23.
8
[9] D. Cvetkovic, I. Gutman and N. Trinajstic, Kekulé structures and topology, Chem. Phys. Lett. 16 (1972) 614–616.
9
[10] D. Cvetkovic, I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. VII. The role of resonance structures, J. Chem. Phys. 61 (1974) 2700–2706.
10
[11] D. Cvetkovic and P. Rowlinson, The largest eigenvalue of a graph: A survey, Linear and Multilinear Algebra 28 (1990) 3–33.
11
[12] D. Cvetkovic, Graphs and their spectra, Publ. Elektrotehn. Fak. (Ser. Mat. Fiz.) 354 (1971) 1–50.
12
[13] D. Cvetkovic and I. Gutman, Note on branching, Croat. Chem. Acta 49 (1977) 115–121.
13
[14] D. Cvetkovic and S. Simic, Graph theoretic results obtained by the support of the expert system “graph”, Bull. Cl. Sci. Math. Nat. Sci. Math. 19 (1994) 19–41.
14
[15] D. Cvetkovic and I. Gutman, The computer system GRAPH: A useful tool in chemical graph theory, J. Comput. Chem. 7 (1986) 640–644.
15
[16] D. Cvetkovic, P. W. Fowler, P. Rowlinson and D. Stevanovic, Constructing fullerene graphs from their eigenvalues and angles, Linear Algebra Appl. 356 (2002) 37–56.
16
[17] D. Cvetkovic and D. Stevanovic, Spectral moments of fullerene graphs, MATCH Commun. Math. Comput. Chem. 50 (2004) 63–72.
17
[18] D. Cvetkovic, Characterizing properties of some graph invariants related to electron charges in the Hückel molecular orbital theory, in: P. Hansen, P. Fowler, M. Zheng (Eds.), Discrete Mathematical Chemistry, American Mathematical
18
Society, Providence, RI, 2000, pp. 79–84.
19
[19] D. Cvetkovic and J. Grout, Graphs with extremal energy should have a small number of distinct eigenvalues, Bull. Cl. Sci. Math. Nat. Sci. Math. 32 (2007) 43–57.
20
[20] D. Cvetkovic, Z. Dražic, V. Kovacevic–Vujcic and M. Cangalovic, The traveling salesman problem: the spectral radius and the length of an optimal tour, Bull. Cl. Sci. Math. Nat. Sci. Math. 43 (2018) 17–26.
21
[21] D. Cvetkovic, Bichromaticity and graph spectrum, Matematicka Biblioteka, (in Serbian) 41 (1969) 193–194.
22
[22] D. Cvetkovic, I. Gutman and N. Trinajstic, Conjugated molecules having integral graph spectra, Chem. Phys. Lett. 29 (1974) 65–68.
23
[23] G. Caporossi, D. Cvetkovic, I. Gutman and P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996.
24
[24] C. A. Coulson, Notes on the secular determinant in molecular orbital theory, Proc. Cambridge Phil. Soc. 46 (1950) 202–205.
25
[25] S. J. Cyvin and I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Springer, Berlin, 1988.
26
[26] I. S. Dmitriev, Molecules without Chemical Bonds?, Himiya, Leningrad (in Russian) 1980, Mir, Moscow (in English) 1981, Deutscher Verlag der Grundstoffindustrie, Leipzig (in German) 1982.
27
[27] A. Graovac, I. Gutman, N. Trinajstic and T. Živkovic, Graph theory and molecular orbitals. Application of Sachs theorem, Theor. Chim. Acta 26 (1972) 67–78.
28
[28] I. Gutman, Impact of the Sachs theorem on theoretical chemistry: A participant’s testimony, MATCH Commun. Math. Comput. Chem. 48 (2003) 17–34.
29
[29] I. Gutman and D. Vidovic, Two early branching indices and the relation between them, Theor. Chem. Acc. 108 (2002) 98–102.
30
[30] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz 103 (1978) 1–22.
31
[31] I. Gutman and B. Furtula, Survey of graph energies, Math. Interdisc. Res. 2 (2017) 85–129.
32
[32] Hs. H. Günthard and H. Primas, Zusammenhang von Graphentheorie und MOâARTheorie von Molekeln mit Systemen konjugierter Bindungen, Helv. Chim. Acta 39 (1956) 1645–1653.
33
[33] W. H. Haemers and Q. Xiang, Strongly regular graphs with parameters (4m4; 2m4+m2;m4+m2;m4+m2) exist for all m > 1, European J. Combin. 31 (2010) 1553–1559.
34
[34] L. Lovász and J. Pelikán, On the eigenvalues of trees, Period. Math. Hungar. 3 (1973) 175–182.
35
[35] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
36
[36] R. B. Mallion and D. H. Rouvray, The golden jubilee of the Coulson– Rushbrooke pairing theorem, J. Math. Chem. 5 (1990) 1–21.
37
[37] H. Sachs, Beziehungen zwischen den in einem Graphen enthaltenen Kreisen und seinem charakteristischen Polynom, Publ. Math. Debrecen 11 (1964) 119–134.
38
ORIGINAL_ARTICLE
Seidel Integral Complete Split Graphs
In the paper we consider a generalized join operation, that is, the H-join on graphs where H is an arbitrary graph. In terms of Seidel matrix of graphs we determine the Seidel spectrum of the graphs obtained by this operation on regular graphs. Some additional consequences regarding S-integral complete split graphs are also obtained, which allows to exhibit many infinite families of Seidel integral complete split graphs.
https://mir.kashanu.ac.ir/article_96006_a1c746b4ddcdc566dcd6fa5e65f3851b.pdf
2019-12-01
137
150
10.22052/mir.2019.194302.1156
Seidel spectrum
Seidel integral graph
H-join of graphs
complete split graph
Pavel
Hic
phic@truni.sk
1
Faculty of Education, Trnava University, Trnava, Slovakia
AUTHOR
Milan
Pokorny
mpokorny@truni.sk
2
Faculty of Education, Trnava University, Trnava, Slovakia
AUTHOR
Dragan
Stevanovic
dragance106@yahoo.com
3
Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia
LEAD_AUTHOR
[1] B. Arsic, D. Cvetkovic, S. K. Simic and M. Škaric, Graph spectral techniques in computer sciences, Appl. Anal. Discrete Math. 6 (2012) 1-30.
1
[2] K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. K. Simic and D. Stevanovic, A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 42-65.
2
[3] D. M. Cardoso, M. A. A. de Freitas, E. A. Martins and M. Robbiano, Spectra of graphs obtained by a generalization of the join graph operation, Discrete Math. 313 (2013) 733-741.
3
[4] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs - Theory and Applications, Third Edition, Johann Ambrosius Barth, Heidelberg, 1995.
4
[5] D. Cvetkovic and S. K. Simic, Towards a spectral theory of graphs based on the signless Laplacian. I., Publ. Inst. Math. (Beograd) (N.S.) 85 (99) (2009) 19-33.
5
[6] M. Fiedler, Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl. 9 (1974) 119–142.
6
[7] M. A. A. de Freitas, N. M. M. de Abreu, R. R. Del-Vecchio and S. Jurkiewicz, Infinite families of Q-integral graphs, Linear Algebra Appl. 432 (2010) 2352-2360.
7
[8] P. Hansen, H. Mélot and D. Stevanovic, Integral complete split graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 89-95.
8
[9] F. Harary and A. J. Schwenk, Which graphs have integral spectra? In: R. A. Bari and F. Harary (Eds.) Graphs and Combinatorics, Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg, 1974.
9
[10] P. Híc and M. Pokorný, Remarks on D-integral complete multipartite graphs, Czechoslovak Math. J. 66 (141)(2) (2016) 457-464.
10
[11] S. Kirkland, M. A. A. de Freitas, R. R. Del Vecchio and N. M. M. de Abreu, Split non-threshold Laplacian integral graphs, Linear Multilinear Algebra 58 (2010) 221-233.
11
[12] S. Lv, The Seidel polynomial and spectrum of the complete 4-partite graphs, J. Northwest Norm. Univ. Nat. Sci. 47 (2) (2011) 22-25.
12
[13] S. Lv, L. Wei and H. Zhao, On the Seidel integral complete multipartite graphs, Acta Math. Appl. Sin. Engl. Ser. 28 (4) (2012) 705-710.
13
[14] R. Merris, Split graphs, Euro. J. Combin. 24 (2003) 413-430.
14
[15] M. Pokorný, QLS integrality of complete r-partite graphs, Filomat 29 (5) (2015) 1043-1051.
15
[16] M. Pokorný, P. Híc and D. Stevanovic, Remarks on Q-integral complete multipartite graphs, Linear Algebra Appl. 439 (2013) 2029-2037.
16
[17] M. Pokorný, P. Híc, D. Stevanovic and M. Miloševic, On distance integral graphs, Discrete Math. 338 (10) (2015) 1784-1792.
17
[18] L. Wang, G. Zhao and Ke. Li, Seidel integral complete r-partite graphs, Graphs Combin. 30 (2) (2014) 479-493.
18
[19] B. F. Wu, Y. Y. Lou and Ch. X. He, Signless Laplacian and normalized Laplacian on the H-join operation of graphs, Discrete Math. Algorithms Appl. 6 (2014) 1450046, 13 pp.
19
[20] N. Zhao and T.Wu, A study of S-integral graphs of complete 5-partite graphs, Pure Appl. Math. 30 (5) (2014) 467-473.
20
[21] N. Zhao, T. Wu and C. Guo, The necessary and sufficient condition for the complete 6-partite graphs to be S-integral, Pure Appl. Math. 29 (2) (2013) 132-139.
21
ORIGINAL_ARTICLE
Oboudi-Type Bounds for Graph Energy
The graph energy is the sum of absolute values of the eigenvalues of the (0, 1)-adjacency matrix. Oboudi recently obtained lower bounds for graph energy, depending on the largest and smallest graph eigenvalue. In this paper, a few more Oboudi-type bounds are deduced.
https://mir.kashanu.ac.ir/article_96938_7a344e0905c77f7e7c5531dd406edc2c.pdf
2019-12-01
151
155
10.22052/mir.2019.207442.1172
Spectral graph theory
Spectrum (of graph)
Graph energy
energy (of graph)
Oboudi-type bounds
Ivan
Gutman
gutman@kg.ac.rs
1
Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia
LEAD_AUTHOR
[1] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs – Theory and Application, Academic Press, New York, 1980; 2nd revised ed.: Barth, Heidelberg, 1995.
1
[2] D. Cvetkovic, P. Rowlinson and S. Simic, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010.
2
[3] I. Gutman, Total -electron energy of benzenoid hydrocarbons, Topics Curr. Chem. 162 (1992) 29–63.
3
[4] I. Gutman and T. Soldatovic, (n,m)-Type approximations for total Φ-electron energy of benzenoid hydrocarbons, MATCH Commun. Math. Comput. Chem. 44 (2001) 169–182.
4
[5] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
5
[6] B. J. McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies, J. Chem. Phys. 54 (1971) 640–643.
6
[7] M. R. Oboudi, A new lower bound for the energy of graphs, Linear Algebra Appl. 590 (2019) 384–395.
7
ORIGINAL_ARTICLE
A Study of PageRank in Undirected Graphs
The PageRank (PR) algorithm is the base of Google search engine. In this paper, we study the PageRank sequence for undirected graphs of order six by PR vector. Then, we provide an ordering for graphs by variance of PR vector which it’s variation is proportional with variance of degree sequence. Finally, we introduce a relation between domination number and PR-variance of graphs.
https://mir.kashanu.ac.ir/article_100994_4a176ba385e4ccacb68137f1ffe36250.pdf
2019-12-01
157
169
10.22052/mir.2018.125190.1097
PageRank algorithm
google matrix
Domination number
isomorphism
Abdollah
Lotfi
math.a.lotfi@gmail.com
1
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran
AUTHOR
Modjtaba
Ghorbani
ghorbani30@gmail.com
2
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran
LEAD_AUTHOR
Hamid
Mesgarani
hmesgarani@srttu.edu
3
Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran
AUTHOR
[1] C. J. Augeri, On Graph Isomorphism and the PageRank Algorithm, Ph.D. Thesis, Air Force Institute of Technology, USA, 2008.
1
[2] P. Berkhin, A survey on PageRank computing, Internet Math. 2 (2005) 73–120.
2
[3] S. Brin and L. Page, Reprint of: The anatomy of a large-scale hypertextual web search engine, Comput. Netw. 56 (2012) 3825–3833.
3
[4] S. Brown, A PageRank model for player performance assessment in basketball, soccer and hockey, arXiv:1704.00583.
4
[5] D. F. Gleich, PageRank beyond the web, SIAM Rev. 57 (2015) 321–363.
5
[6] V. Grolmusz, A note on the PageRank of undirected graphs, Inform. Process. Lett. 115 (2012) 633–634.
6
[7] B. Jiang, K. kloster, D. F. Gleich and M. Gribskov, AptRank: an adaptive PageRank model for protein function prediction on bi-relational graphs, Bioinformatics 33 (2017) 1829–1836.
7
[8] V. Kandiah and D. L. Shepelyansky, PageRank model of opinion formation on social networks, Physica A 391 (2012) 5779–5793.
8
[9] A. N. Langville and C. D. Meyer, Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press, Princeton and Oxford, 2005.
9
[10] V. Lazova and L. Basnarkov, PageRank approach to ranking national football teams, arXiv:1503.01331 (2015).
10
[11] Ch. Liao, K. Lu, M. Baym, R. Singh and B. Berger, IsoRankn: spectral methods for global alignment of multiple protein networks, Bioinformatics 25 (2009) i253–i258.
11
[12] B. L. Mooney, L. R. Corrales and A. E. Clark, Molecularnetworks: An integrated graph theoretic and data mining tool to explore solvent organization in molecular simulation, J. Comput. Chem. 33 (2012) 853–860.
12
[13] J. L. Morrison, R. Breitling, D. J. Higham and D. R. Gilbert, GeneRank: Using search engine technology for the analysis of microarray experiments, BMC Bioinform. DOI: 10.1186/1471-2105-6-233.
13
[14] N. Mukai, PageRank-based traffic simulation using taxi probe data, Procedia Comput. Sci. 22 (2013) 1156–1163.
14
[15] F. Pedroche, E. García, M. Romance and R. Criado, Sharp estimates for the personalized multiplex PageRank, J. Comput. Appl. Math. 330 (2018) 1030–1040.
15
[16] Z. L. Shen, T. Z. Huang, B. Carpentieri, X. Gu and C. Wen, An efficient elimination strategy for solving PageRank problem, Appl. Math. Comput. 298 (2017) 111–122.
16
[17] C. Wen, T. Z. Huang and Z. L. Shen, A note on the two-step matrix splitting iteration for computing PageRank, J. Comput. Appl. Math. 315 (2017) 87–97.
17
[18] Ch. Winter, G. Kristiansen, S. Kersting, J. Roy, D. Aust, T. Knöse, P. Rümmele, B. Jahnke, V. Hentrich, F. Rückert, M. Niedergethmann, W. Weichert and M. Bahra, Google goes cancer: Improving outcome prediction for cancer patients by network-based ranking of marker genes, PLOS Comput. Biol. 2 (2012) e1002511.
18
ORIGINAL_ARTICLE
Note on the Sum of Powers of Normalized Signless Laplacian Eigenvalues of Graphs
In this paper, for a connected graph G and a real α≠0, we define a new graph invariant σα(G)-as the sum of the alphath powers of the normalized signless Laplacian eigenvalues of G. Note that σ1/2(G) is equal to Randic (normalized) incidence energy which have been recently studied in the literature [5, 15]. We present some bounds on σα(G) (α ≠ 0, 1) and also consider the special case α = 1/2.
https://mir.kashanu.ac.ir/article_101587_6d3f9f9d05078067f97d041c27644362.pdf
2019-12-01
171
182
10.22052/mir.2019.208991.1180
Normalized signless Laplacian eigenvalues
Randic (normalized) incidence energy
Bound
Ş. Burcu
Bozkurt Altındağ
srf_burcu_bozkurt@hotmail.com
1
Konya, Turkey
LEAD_AUTHOR
[1] M. Bianchi, A. Cornaro, J. L. Palacios and A. Torriero, Bounding the sum of powers of normalized Laplacian eigenvalues of graphs through majorization methods, MATCH Commun. Math. Comput. Chem. 70 (2013) 707–716.
1
[2] S. B. Bozkurt, A. D. Gungor, I. Gutman and A. S. Cevik, Randic matrix and Randic energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 321–334.
2
[3] S. B. Bozkurt and D. Bozkurt, On the sum of powers of normalized Laplacian eigenvalues of graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 917–930.
3
[4] S. B. Bozkurt and I. Gutman, Estimating the incidence energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 143–156.
4
[5] B. Cheng and B. Liu, The normalized incidence energy of a graph, Linear Algebra Appl. 438 (2013) 4510–4519.
5
[6] H. Chen and F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (2007) 654–661.
6
[7] F. R. K. Chung, Spectral Graph Theory, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1997.
7
[8] G. P. Clemente and A. Cornaro, New bounds for the sum of powers of normalized Laplacian eigenvalues of graphs, Ars Math. Contemp. 11 (2016) 403–413.
8
[9] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Academic press, New York, 1980.
9
[10] D. Cvetkovic, P. Rowlinson and S. Simic, Signless Laplacian of finite graphs, Linear Algebra Appl. 423 (2007) 155–171.
10
[11] D. Cvetkovic and S. Simic, Towards a spectral theory of graphs based on the signless Laplacian, I, Publ. Inst. Math. (Beograd) 85 (2009) 19–33.
11
[12] D. Cvetkovic and S. Simic, Towards a spectral theory of graphs based on the signless Laplacian, II, Linear Algebra Appl. 432 (2010) 2257–2277.
12
[13] D. Cvetkovic and S. Simic, Towards a spectral theory of graphs based on the signless Laplacian, III, Appl. Anal. Discrete Math. 4 (2010) 156–166.
13
[14] K. Ch. Das, A. D. Gungor and S. B. Bozkurt, On the normalized Laplacian eigenvalues of graphs, Ars Combin. 118 (2015) 143–154.
14
[15] R. Gu, F. Huang and X. Li, Randic incidence energy of graphs, Trans. Comb. 3 (2014) 1–9.
15
[16] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz 103 (1978) 1–22.
16
[17] I. Gutman, The McClelland approximation and the distribution of π-electron molecular orbital energy levels, J. Serb. Chem. Soc. 72 (2007) 967–973.
17
[18] I. Gutman, A. V. Teodorovic and Lj. Nedeljkovic, Topological properties of benzenoid systems. Bounds and approximate formula for total -electron energy, Theor. Chem. Acc. 65 (1984) 23–31.
18
[19] I. Gutman, D. Kiani and M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009) 573–580.
19
[20] I. Gutman, D. Kiani, M. Mirzakhah and B. Zhou, On incidence energy of a graph, Linear Algebra Appl. 431 (2009) 1223–1233.
20
[21] I. Gutman, B. Zhou and B. Furtula, The Laplacian-energy like invariant is an energy like invariant, MATCH Commun. Math. Comput. Chem. 64 (2010) 85–96.
21
[22] R. Jooyandeh, D. Kiani and M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem. 62 (2009) 561–572.
22
[23] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
23
[24] J. Li, J. M. Guo, W. C. Shiu, S. B. Bozkurt Altındag and D. Bozkurt, Bounding the sum of powers of normalized Laplacian eigenvalues of a graph, Appl. Math. Comput. 324 (2018) 82–92.
24
[25] B. Liu, Y. Huang and J. Feng, A note on the Randic spectral radius, MATCH Commun. Math. Comput. Chem. 68 (2012) 913–916.
25
[26] B. Liu, Y. Huang and Z. You, A survey on the Laplacian-energy like invariant, MATCH Commun. Math. Comput. Chem. 66 (2011) 713–730.
26
[27] J. Liu and B. Liu, A Laplacian-energy-like invariant of a graphs, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372.
27
[28] R. Merris, Laplacian matrices of graphs A survey, Linear Algebra Appl. 197-198 (1994) 143–176.
28
[29] R. Merris, A survey of graph Laplacians, Linear and Multilinear Algebra 39 (1995) 19–31.
29
[30] E. I. Milovanovic, M. M. Matejic and I. Ž. Milovanovic, On the normalized Laplacian spectral radius, Laplacian incidence energy and Kemeny’s constant, Linear Algebra Appl. 582 (2019) 181–196.
30
[31] D. S. Mitrinovic and P. M. Vasic, Analytic Inequalities, Springer-Verlag, New York,1970, pp.74–94.
31
[32] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007) 1472–1475.
32
[33] L. Shi, H. Wang, The Laplacian incidence energy of graphs, Linear Algebra Appl. 439 (2013) 4056–4062.
33
[34] B. Zhou, I. Gutman and T. Aleksic, A note on the Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441–446.
34
[35] B. Zhou, More upper bounds for the incidence energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 123–128.
35
ORIGINAL_ARTICLE
The Fourth and Fifth Laplacian Coefficients of some Rooted Trees
Abstract. The Laplacian characteristic polynomial of an n-vertex graph G has the form f(G,x) = xn+∑lixn-i. In this paper, the fourth and fifth coefficient of f(G,x), will be investigated, where G is a T(k,t) tree in which a rooted tree with degree sequence k,k,...,k,1,1,...,1 is denoted by T(k,t).
https://mir.kashanu.ac.ir/article_101588_f54ef31c99328589cc826b00c2bd8846.pdf
2019-12-01
183
192
10.22052/mir.2020.207378.1173
Graph
Eigenvalue
Laplacian matrix
Laplacian coefficient
Mahsa
Arabzadeh
mahsa.arabzade1177@gmail.com
1
Department of Mathematics, Islamic Azad University, Science and Researcher Branch Tehran, I. R. Iran
AUTHOR
Gholam-Hossein
Fath-Tabar
fathtabar@kashanu.ac.ir
2
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran
LEAD_AUTHOR
Hamid
Rasoli
hrasouli@srbiau.ac.ir
3
Department of Mathematics, Islamic Azad University, Science and Researcher Branch Tehran, I. R. Iran
AUTHOR
Abolfazl
Tehranian
tehranian@srbiau.ac.ir
4
Department of Mathematics, Islamic Azad University, Science and Researcher Branch Tehran, I. R. Iran
AUTHOR
[1] A. R. Ashrafi, M. Eliasi and A. Ghalavand, Laplacian coefficients and Zagreb indices of trees, Linear Multilinear Algebra 67 (2019) 1736–1749.
1
[2] N. Biggs, Algebraic Graph Theory, Cambridge Univ Press, Cambridge, 1993.
2
[3] A. Behmaram, On the number of 4-matchings in graphs, MATCH Commun. Math. Comput. Chem. 62 (2009) 381–388.
3
[4] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra in Graph–Theory and Application, Academic Press, New York, 1980.
4
[5] G. H. Fath-Tabar, A. R. Ashrafi and I. Gutman, Note on Estrada and LEstrada indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math. 139 (2009) 1–16.
5
[6] G. H. Fath-Tabar, T. Došlic and A. R. Ashrafi, On the Szeged and the Laplacian Szeged spectrum of a graph, Linear Algebra Appl. 433 (2010) 662–671.
6
[7] G. H. Fath-Tabar and A. R. Ashrafi, Some remarks on Laplacian eigenvalues and Laplacian energy of graphs, Math. Commun. 15 (2010) 443–451.
7
[8] E. J. Farrel, J. M. Guo and G. M. Constantine, On matching coefficients, Discrete Math. 89 (1991) 203–210.
8
[9] I. Gutman and L. Pavlovic, On the coefficients of the Laplacian characteristic polynomial of trees, Bull. Cl. Sci. Math. Nat. Sci. Math. 28 (2003) 31–40.
9
[10] C. S. Oliveira, N. M. Maia de Abreu and S. Jurkiewicz, The characteristic polynomial of the Laplacian of graphs in (a; b)-linear classes, Special issue on algebraic graph theory, Linear Algebra Appl. 356 (2002) 113–121.
10
[11] F. Taghvaee and G. H. Fath-Tabar, On the skew spectral moments of graphs, Trans. Comb. 6 (2017) 47–54.
11
[12] F. Taghvaee and G. H. Fath-Tabar, Relationship between coefficients of characteristic polynomial and matching polynomial of regular graphs and its applications, Iranian J. Math. Chem. 8 (2017) 7–23.
12
ORIGINAL_ARTICLE
A Multiplicative Version of Forgotten Topological Index
In this paper, we present upper bounds for the multiplicative forgotten topological index of several graph operations such as sum, Cartesian product, corona product, composition, strong product, disjunction and symmetric difference in terms of the F–index and the first Zagreb index of their components. Also, we give explicit formulas for this new graph invariant under two graph operations such as union and Tensor product. Moreover, we obtain the expressions for this new graph invariant of subdivision graphs and vertex – semitotal graphs. Finally, we compare the discriminating ability of indices.
https://mir.kashanu.ac.ir/article_102000_e2b2c1124183c3b89ccf0bc81f103cf6.pdf
2019-12-01
193
211
10.22052/mir.2019.176557.1126
topological index
multiplicative forgotten topological index
Graph operations
subdivision graphs
vertex – semitotal graphs
Asghar
Yousefi
naser.yosefi53@yahoo.com
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
AUTHOR
Ali
Iranmanesh
iranmana@yahoo.com
2
Department of Mathematics, Tarbiat Modares University, Tehran, Iran
LEAD_AUTHOR
Andrey
Dobrynin
dobr@math.nsc.ru
3
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
AUTHOR
Abolfazl
Tehranian
tehranian@srbiau.ac.ir
4
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
AUTHOR
[1] A. R. Ashrafi, T. Doslic and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571–1578.
1
[2] M. Azari and A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013) 901–919.
2
[3] M. Azari and A. Iranmanesh, Computation of the edge Wiener indices of the sum of graphs, Ars Combin. 100 (2011) 113–128.
3
[4] M. Azari and A. Iranmanesh, Computing the eccentric– distance sum for graph operations, Discrete Appl. Math. 161 (2013) 2827–2840.
4
[5] M. Azari and A. Iranmanesh, Some inequalities for the multiplicative sum Zagreb index of graph operations, J. Math. Inequal. 9 (2015) 727–738.
5
[6] M. Azari, A. Iranmanesh and I. Gutman, Zagreb indices of bridge and chain graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 921–938.
6
[7] M. Azari, A. Iranmanesh and A. Tehranian, A method for calculating an edge version of the Wiener number of a graph operation, Util. Math. 87 (2012) 151–164.
7
[8] B. Basavanagoud and V. R. Desai, Forgotten topological index and hyper–Zagreb index of generalized transformation graphs, Bull. Math. Sci. Appl. 14 (2016) 1–6.
8
[9] B. Basavanagoud, I. Gutman and C. S. Gali, On second Zagreb index and coindex of some derived graphs, Kragujevac J. Sci. 37 (2015) 113–121.
9
[10] N. De, S. M. A. Nayeem and A. Pal, F-index of some graph operations,Discrete Math. Algorithms Appl. 8 (2016) 1650025, 17 pp.
10
[11] N. De, S. M. A. Nayeem and A. Pal, Reformulated first Zagreb index of some graph operations, Mathematics 3 (2015) 945-960.
11
[12] J. Devillers and A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999.
12
[13] M. V. Diudea, QSPR/ QSAR Studies by Molecular Descriptors, Nova Sci. Publ., Huntington, NY, 2000.
13
[14] T. Došlic, B. Furtula, A. Graovac, I. Gutman, S. Moradi and Z. Yarahmadi, On vertex – degree – based molecular structure descriptors, MATCH Commun. Math. Comput. Chem. 66 (2011) 613–626.
14
[15] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184–1190.
15
[16] B. Furtula, I. Gutman, Z. Kovijanić Vuki´cević, G. Lekishvili and G. Popivoda,On an old/new degree – based topological index, Bulletin T.CXLVIII de l' Academie serbe des sciences et des arts (2015) 19–31.
16
[17] W. Gao, M. R. Farahani and L. Shi, The forgotten topological index of some drug structures, Acta Medica Mediterranea 32 (2016) 579–585.
17
[18] S. Ghobadi and M. Ghorbaninejad, The forgotten topological index of four operations on some special graphs, Bulletin of Mathematical Sciences and Applications 16 (2016) 89–95.
18
[19] I. Gutman, B. Furtula, Zˇ. Kovijanić Vuki´cević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015) 5–16.
19
[20] I. Gutman, B. Rušcic, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 33–99.
20
[21] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π–electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
21
[22] Y. Hu, X. Li, Y. Shi, T. Xu and I. Gutman, On molecular graphs with smallest and greatest zeroth–order general Randic index, MATCH Commun. Math. Comput. Chem. 54 (2005) 425–434.
22
[23] M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley–Interscience, New York, 2000.
23
[24] M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.
24
[25] X. Li and H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004) 57–62.
25
[26] X. Li and J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 195–208.
26
[27] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124.
27
[28] K. Pattabiraman and P. Kandan, Weighted PI index of corona product of graphs, Discret. Math. Algorithms Appl. 6 (2014) 1450055, 9 pp.
28
[29] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley–VCH, Weinheim, 2000.
29
[30] M. Wang and H. Hua, More on Zagreb coindices of composite graphs, Int. Math. Forum 7 (2012) 669–673.
30
[31] Z. Yarahmadi and A. R. Ashrafi, The Szeged, vertex PI, first and second Zagreb indices of corona product of graphs, Filomat 26 (2012) 467–472.
31
[32] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004) 113–118.
32
ORIGINAL_ARTICLE
On the Configurations with n Points and Two Distances
In this paper we investigate the geometric structures of M(n, 2) containing n points in R^3 having two distinct distances. We will show that up to pseudo-equivalence there are 5 constructible models for M(4, 2) and 17 constructible models for M(5, 2).
https://mir.kashanu.ac.ir/article_45816_fa687f17a0883ef1c1290ad251cdd442.pdf
2019-12-01
213
225
10.22052/mir.2017.81496.1056
Constructible models
distinct distances
isomorphic graphs
pseudo-equivalent models
Ali Asghar
Rezaei
a_rezaei@kashanu.ac.ir
1
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran
LEAD_AUTHOR
[1] D. Avis, P. Erdös and J. Pach, Distinct distances determined by subsets of a point set in space, Comput. Geom. 1 (1991) 1 - 11.
1
[2] F. R. K. Chung, The number of different distances determined by n points in the plane, J. Combin. Theory Ser. A 36 (1984) 342 - 354.
2
[3] F. R. K. Chung, E. Szemerédi and W. T. Trotter, The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7 (1992) 1 - 11.
3
[4] H. T. Croft, 9-point and 7-point configurations in 3-space, Proc. London Math. Soc. 12 (1962) 400 - 424.
4
[5] P. Erdös, On sets of distances of n points, Amer. Math. Monthly 53 (1946) 248 - 250.
5
[6] P. Erdos and G. Purdy, Some extremal problems in geometry, J. Combinatorial Theory Ser. A 10 (1971) 246 - 252.
6
[7] L. Moser, On the different distances determined by n points, Amer. Math. Monthly 59 (1952) 85 - 91.
7
[8] A. A. Rezaei, On the geometric structures with n points and k distances, Electronic Notes Discrete Math. 45 (2014) 181 - 186.
8
ORIGINAL_ARTICLE
Trees with Extreme Values of Second Zagreb Index and Coindex
In this paper we present a generalization of the aforementioned bound for all trees in terms of the order and maximum degree. We also give a lower bound on the second Zagreb coindex of trees.
https://mir.kashanu.ac.ir/article_64769_94f1d4167b1f10a85ce4b7a462e66070.pdf
2019-12-01
227
238
10.22052/mir.2018.130441.1100
Zagreb index
second Zagreb index
second Zagreb coindex
Tree
Reza
Rasi
r.rasi@azaruniv.ac.ir
1
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I. R. Iran
AUTHOR
Seyed Mahmoud
Sheikholeslami
sm.sheikholeslami@azaruniv.edu
2
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I. R. Iran
AUTHOR
Afshin
Behmaram
behmarammath@gmail.com
3
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, I. R. Iran
LEAD_AUTHOR
[1] A. R. Ashrafi, T. Došlić and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571–1578.
1
[2] K. Ch. Das, Sharp bounds for the sum of the squares of the degrees of a graph, Kragujevac J. Math. 25 (2003) 31–49.
2
[3] K. Ch. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285 (2004) 57–66.
3
[4] D. de Caen, An upper bound on the sum of squares in a graph, Discrete Math. 185 (1998) 245–248.
4
[5] T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008) 66–80.
5
[6] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals, total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
6
[7] Ž. Kovijanić Vukićević and G. Popivoda, Chemical trees with extreme values of Zagreb indices and coindices, Iranian J. Math. Chem. 5 (2014) 19–29.
7
[8] X. Li and I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, Mathematical Chemistry Monograph 1, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, 2006.
8
[9] S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124
9
[10] R. Rasi, S. M. Sheikholeslami and A. Behmaram, An upper bound on the first Zagreb index in trees, Iranian J. Math. Chem. 8 (1) (2017) 71–82.
10
[11] S. Zhang, W. Wang and T. C. E. Cheng, Bicyclic graphs with the first three smalllest and largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem. 56 (2006) 579–592.
11
[12] B. Zhou and I. Gutman, Relations betweenWiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004) 93–95.
12
ORIGINAL_ARTICLE
Distinguishing Number and Distinguishing Index of the Join of Two Graphs
The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let Gk be the k-th power of G with respect to the join product. We prove that if G is a connected graph of order n ≥ 2, then Gk has the distinguishing index 2, except D'(K2+K2)=3.
https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdf
2019-12-01
239
251
10.22052/mir.2020.133523.1102
Distinguishing index
distinguishing number
join
Saeid
Alikhani
alikhani206@gmail.com
1
Department of Mathematics, Yazd University, Yazd, Iran
LEAD_AUTHOR
Samaneh
Soltani
s.soltani1979@gmail.com
2
Department of Mathematics, Yazd University, Yazd, Iran
AUTHOR
[1] M. O. Albertson, Distinguishing Cartesian powers of graphs, Electron. J. Combin. 12 (2005) #N17, 5 pp.
1
[2] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996) #R18, 1 - 17.
2
[3] S. Alikhani and S. Soltani, Distinguishing number and distinguishing index of certain graphs, Filomat 31 (2017) 4393 - 4404.
3
[4] B. Bogstad and L. J. Cowen, The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29 - 35.
4
[5] M. J. Fisher and G. Isaak, Distinguishing colorings of Cartesian products of complete graphs, Discrete Math. 308 (2008) 2240 - 2246.
5
[6] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, Chapman and Hall/CRC, 2011.
6
[7] W. Imrich, J. Jerebic and S. Klavžar, The distinguishing number of Cartesian products of complete graphs, European J. Combin. 29 (2008) 922 - 929.
7
[8] R. Kalinowski and M. Pilsniak, Distinguishing graphs by edge-colourings, European J. Combin. 45 (2015) 124 - 131.
8
[9] S. Klavžar and X. Zhu, Cartesian powers of graphs can be distinguished by two labels, European J. Combin. 28 (2007) 303 - 310.
9
[10] M. Pilsniak, Improving upper bounds for the distinguishing index, Ars Math. Contemp. 13 (2017) 259 - 274.
10
ORIGINAL_ARTICLE
Probabilistic Properties of F-indices of Trees
The aim of this paper is to introduce some results for the F-index of the tree structures without any information on the exact values of vertex degrees. Three martingales related to the first Zagreb index and F-index are given.
https://mir.kashanu.ac.ir/article_102110_03ce4c5cdfca7c2b30cafd5a8b6251ba.pdf
2019-12-01
253
261
10.22052/mir.2019.183327.1130
Tree structures
F-indices
martingale
Hadis
Morovati
rst.kazemi@gmail.com
1
Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran
AUTHOR
Ramin
Kazemi
r.kazemi@sci.ikiu.ac.ir
2
Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran
LEAD_AUTHOR
Akram
Kohansal
kazemi@ikiu.ac.ir
3
Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran
AUTHOR
[1] P. Billingsley, Probability and Measure, John Wiley & Sons, New York, 1885.
1
[2] Z. Che and Z. Chen, Lower and upper bounds of the forgotten topological index, MATCH Commun. Math. Comput. Chem. 76 (2016) 635–648.
2
[3] T. Doštlic, T. Réti and D. Vukicevic, On the vertex degree indices of connected graphs, Chem. Phys. Lett. 512 (2011) 283–286.
3
[4] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184–1190.
4
[5] W. Gao, M. R. Farahani and L. Shi, Forgotten topological index of some drug structures, Acta Med. Medit. 32 (2016) 579–585.
5
[6] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
6
[7] R. Kazemi, The eccentric connectivity index of bucket recursive trees, Iranian J. Math. Chem. 5 (2014) 77–83.
7
[8] Y. C. Sun, Z. Lin, W. X. Peng, T. Q. Yuan, F. Xu, Y. Q. Wu, J. Yang, Y. S. Wang and R. C. Sun, Chemical changes of raw materials and manufactured binderless boards during hot pressing: lignin isolation and characterization,
8
BioResources 9 (2014) 1055–1071.
9
ORIGINAL_ARTICLE
Classification of Bounded Travelling Wave Solutions of the General Burgers-Boussinesq Equation
By using bifurcation theory of planar dynamical systems, we classify all bounded travelling wave solutions of the general Burgers-Boussinesq equation, and we give their corresponding phase portraits. In different parametric regions, different types of trav- elling wave solutions such as solitary wave solutions, cusp solitary wave solutions, kink(anti kink) wave solutions and periodic wave solutions are simulated. Also in each parameter bifurcation sets, we obtain the exact explicit parametric representation of all travelling wave solutions.
https://mir.kashanu.ac.ir/article_33673_c46440a9ce390b0837c9ce8d88d78e80.pdf
2019-12-01
263
279
10.22052/mir.2016.33673
General Burgers-Boussinesq equation
travelling wave solutions
bifurcation theory
Rasool
Kazemi
r.kazemi@kashanu.ac.ir
1
Department of Pure Mathematics, University of Kashan, Kashan, I. R. Iran
LEAD_AUTHOR
Masoud
Mossadeghi
m.mosaddeghi@math.iut.ac.ir
2
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111
AUTHOR
[1] P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer–Verlag, New York-Heidelberg, 1971.
1
[2] En. G. Fan, Integrable Systems and Computer Algebra, Science Press, 2004.
2
[3] E. Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22.
3
[4] D. Feng, J. Lu, J. Li and T. He, Bifurcation studies on travelling wave solutions for nonlinear intensity Klein-Gordon equation, Appl. Math. Comput. 189 (2007) 271–284.
4
[5] J. K. Hale and H. Kocak, Dynamics and Bifurcation, Springer–Verlag, New York, 1991.
5
[6] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals 26 (2005) 695–700.
6
[7] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194.
7
[8] J. Q. Hu, An algebraic method exactly solving two high–dimensional nonlinear evolution equations, Chaos, Solitons & Fractals 23 (2005) 391–398.
8
[9] B. Jiang, Y. Lu, J. Zhang and Q. Bi, Bifurcations and some new traveling wave solutions for the CH-γ equation, Appl. Math. Comput. 228 (2014) 220–233.
9
[10] M. Khalfallah, Exact traveling wave solutions of the Boussinesq–Burger equation, Math. Comput. Modelling 49 (2009) 666–671.
10
[11] J. B. Li and Y. S. Li, Bifurcations of travelling wave solutions for a two–component Camassa–Holm equation, Acta Math. Sin. (Engl. Ser.) 24 (2008) 1319–1330.
11
[12] Sh. Liu, Z. Fu, Sh. Liu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 285 (2001) 69–74.
12
[13] S. Y. Lou and X. Y. Tang, Nonlinear Mathematical and Physical Methods, Science Press, 2006.
13
[14] R. M. Miura, Backlund Transformation, the Inverse Scattering Method, Solitons, and their Applications, Springer-Verlage, Berlin, 1976.
14
[15] A. S. A. Rady and M. Khalfallah, On soliton solutions for Boussinesq-Burgers equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 886–894.
15
[16] M. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996) 279–287.
16
[17] J. Weiss, M. Tabor and G. Carnevale, The painleve property for partial differential equations, J. Math. Phys. 24 (1983) 522-526.
17
[18] Ch. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84.
18
[19] H. R. Z. Zangeneh, R. Kazemi and M. Mosaddeghi, Classification of bounded travelling wave solutions of the generalized Zakharov equation, Iran. J. Sci. Technol. Trans. A Sci. 38 (2014) 355–364.
19
[20] K. Zhang and J. Han, Bifurcations of traveling wave solutions for the (2 + 1)–dimensional generalized asymmetric Nizhnik-Novikov-Veselov equation, Appl.Math. Comput. 251 (2015) 108–117.
20
ORIGINAL_ARTICLE
Graph Invariants of Deleted Lexicographic Product of Graphs
The deleted lexicographic product G[H]-nG of graphs G and H is a graph with vertex set V(G)×V(H) and u=(u1, v1) is adjacent with v=(u2, v2) whenever (u1=u2 and v1 is adjacent with v2) or (v1 ≠ v2 and u1 is adjacent with u2). In this paper, we compute the exact values of the Wiener, vertex PI and Zagreb indices of deleted lexicographic product of graphs. Applications of our results under some examples are presented.
https://mir.kashanu.ac.ir/article_102486_8728c06bbd29a7f7c9f2daefaaa2fa55.pdf
2019-12-01
281
291
10.22052/mir.2019.176548.1125
Deleted lexicographic product
Wiener index
Vertex PI index
Zagreb indices
Bahare
Akhavan Mahdavi
bahare.akhavan@um.ac.ir
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, I. R. Iran
AUTHOR
Mostafa
Tavakoli
m_tavakoli@um.ac.ir
2
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, I. R. Iran
AUTHOR
Freydoon
Rahbarnia
rahbarnia@um.ac.ir
3
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, I. R. Iran
AUTHOR
[1] A. R. Ashrafi, A. Karbasioun and M. V. Diudea, Computing Wiener and detour indices of a new type of nanostar dendrimers, MATCH Commun. Math. Comput. Chem. 65 (2011) 193 - 200.
1
[2] M. V. Diudea, Polyhex Tori Originating in Square Tiled Tori, In: M. V. Diudea (Ed.), Nanostructures: Novel Architecture, Nova Science Publishers, New York, 2005, 111 - 126.
2
[3] M. V. Diudea, M. Stefu, B. Pârv and P. E. John, Wiener index of armchair polyhex nanotubes, Croat. Chem. Acta 77 (2004) 111 - 115.
3
[4] M. V. Diudea and I. Gutman, Wiener-type topological indices, Croat. Chem. Acta 71 (1998) 21 - 51.
4
[5] J. Feigenbaum and A. A. Schäffer, Recognizing composite graphs is equivalent to testing graph isomorphism, SIAM J. Comput. 15 (1986) 619 - 627.
5
[6] B. Frelih and Š. Miklavic, Edge regular graph products, Electron. J. Combin. 20 (2013) Paper 62, 17 pp.
6
[7] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535 - 538.
7
[8] I. Gutman, B. Ruscic, N. Trinajstic and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399 - 3405.
8
[9] F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, German, 1914.
9
[10] P. V. Khadikar, S. Karmarkar and V. K. Agrawal, A novel PI index and its applications to QSPR/QSAR studies, J. Chem. Inf. Comput. Sci. 41 (2001) 934 - 949.
10
[11] M. K. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math. 156 (2008) 1780 - 1789.
11
[12] M. K. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804 - 811.
12
[13] M. Tavakoli, F. Rahbarnia and A. R. Ashrafi, Applications of generalized hierarchical product of graphs in computing the vertex and edge PI indices of chemical, Ric. Math. 63 (2014) 59 - 65.
13
[14] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17 - 20.
14
[15] Y. N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (1994) 359 - 365.
15
ORIGINAL_ARTICLE
Best Proximity Point Theorems for Ciric Type G-Contractions in Metric Spaces with a Graph
In this paper, we aim to introduce Ciric type G-contractions using directed graphs in metric spaces and then to investigate the existence and uniqueness of best proximity points for them. We also discuss the main theorem and list some consequences of it.
https://mir.kashanu.ac.ir/article_102487_bffa611422018a74ee8545663dd0461f.pdf
2019-12-01
293
304
10.22052/mir.2019.187067.1135
G-proximal mapping
Ciric type G-contraction
Best proximity point
Kamal
Fallahi
fallahi1361@gmail.com
1
Department of Mathematics, Payam Noor University, Tehran, Iran
LEAD_AUTHOR
Mohammad
Hamidi
m.hamidi@pnu.ac.ir
2
Department of Mathematics, Payam Noor University, Tehran, Iran
AUTHOR
[1] S. M. A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Some fixed point results on a metric space with a graph, Topology Appl. 159 (2012) 659-663.
1
[2] M. I. Ayari, Best proximity point theorems for generalized--proximal quasicontractive mappings, Fixed Point Theory Appl. 2017 (2017) 13 pp.
2
[3] M. I. Ayari, M. Berzig and I. Kédim, Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation, Math. Sci. 10 (2016) 105 - 114.
3
[4] S. S. Basha, Discrete optimization in partially ordered sets, J. Global Optim. 54 (2012) 511 - 517.
4
[5] S. Basha, Best proximity point theorems in the frameworks of fairly and proximally complete spaces, J. Fixed Point Theory Appl. 19 (2017) 1939 - 1951.
5
[6] S. S. Basha, N. Shahzad and C. Vetro, Best proximity point theorems for proximal cyclic contractions, J. Fixed Point Theory Appl. 19 (2017) 2647 - 2661.
6
[7] F. Bojor, Fixed points of Kannan mappings in metric spaces endowed with a graph, An. Stiinµ. Univ. “Ovidius" Constanµa Ser. Mat. 20 (2012) 31 - 40.
7
[8] F. Bojor, Fixed point theorems for Reich type contractions on a metric spaces with a graph, Nonlinear Anal. 75 (2012) 3895 - 3901.
8
[9] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
9
[10] C. Chifu and G. Petrusel, Generalized contractions in metric spaces endowed with a graph, J. Fixed Point Theory Appl. 2012 (2012) 9 pp.
10
[11] Lj. B. Ciric, On contraction type mappings, Math. Balkanica. 1 (1971) 52-57.
11
[12] Lj. B. Ciric, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974) 267 - 273.
12
[13] M. Gabeleh and N. Shahzad, Best proximity points, cyclic Kannan maps and geodesic metric spaces, J. Fixed Point Theory Appl. 18 (2016) 167 - 188.
13
[14] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008) 1359 - 1373.
14
[15] V. S. Raj, A best proximity point theorem for weakly contractive non-selfmappings, Nonlinear Anal. 74 (2011) 4804 - 4808.
15
[16] V. S. Raj, Best proximity point theorems for non-self mappings, J. Fixed Point Theory Appl. 14 (2013) 447 - 454.
16
[17] A. Sultana and V. Vetrivel, Best proximity points of contractive mappings on a metric space with a graph and applications, Appl. Gen. Topol. 18 (2017) 13 - 21.
17
ORIGINAL_ARTICLE
k-Intersection Graph of a Finite Set
For any nonempty set Ω and k-subset Λ, the k-intersection graph, denoted by Γm(Ω,Λ), is an undirected simple graph whose vertices are all m-subsets of Ω and two distinct vertices A and B are adjacent if and only if A∩B ⊈ Λ. In this paper, we determine diameter, girth, some numerical invariants and planarity, Hamiltonian and perfect matching of these graphs. ﬁnally we investigate their adjacency matrices.
https://mir.kashanu.ac.ir/article_102613_b205b739f72772023b0d554c0ed5cdc2.pdf
2019-12-01
305
317
10.22052/mir.2020.208185.1178
intersection graph
k-intersection graph
diameter
Fahimeh
Esmaeeli
fahimeh.smaily@gmail.com
1
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, I. R. Iran
AUTHOR
Ahmad
Erfanian
erfanian@um.ac.ir
2
Department of Pure Mathematics and The Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, I. R. Iran
LEAD_AUTHOR
Farzaneh
Mansoori
mansoori.farzaneh@gmail.com
3
Department of Pure Mathematics, Ferdowsi University of Mashhad, International Campus Mashhad, I. R. Iran
AUTHOR
[1] R. B. Bapat, Graphs and Matrices, Springer-Verlag, London, 2010.
1
[2] J. Bosák, The graphs of semigroups, Proc. Symposium Smolenice, Prague (1964) 119–125.
2
[3] B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969) 241–247.
3
[4] I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of ring, Discrete Math. 309 (2009) 5381–5392.
4
[5] L. E. Dickson, History of the Theory of Numbers, Chelsea, 1952.
5
[6] I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Math. 307 (2007) 854–865.
6
[7] C. Godsil and G. F. Royle, Algebraic Graph Theory, Springer, 2001.
7
[8] V. P. Korzhik, Minimal non-1-planar graphs, Discrete Math. 308 (2008) 1319–1327.
8
[9] B. Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25 (100) (1975) 171–174.
9
ORIGINAL_ARTICLE
On Eigenvalues of Permutation Graphs
Let λ1(G), λ2(G),..., λs(G) be the distinct eigenvalues of G with multiplicities t1, t2,..., ts, respectively. The multiset {λ1(G)t1, λ2(G)t2,..., λs(G)ts} of eigenvalues of A(G) is called the spectrum of G. For two graphs G and H, if their spectrum are the same, then G and H are said to be co-spectral. The aim of this paper is to determine co-spectral permutation graphs with respect to automorphism group of graph G.
https://mir.kashanu.ac.ir/article_102948_a93fee635635058ad0063521ba55abff.pdf
2019-12-01
319
325
10.22052/mir.2020.213088.1189
Permutation graph
Petersen graph
Automorphism group
Sima
Saadat-Akhtar
simasaadatzzz3@gmail.com
1
Department of Mathematics, Faculty of Basic Sciences, Islamic Azad University, Central Tehran Branch, Tehran, Iran
AUTHOR
Shervin
Sahebi
sahebi@iauctb.ac.ir
2
Department of Mathematics, Faculty of Basic Sciences, Islamic Azad University, Central Tehran Branch, Tehran, Iran
LEAD_AUTHOR
[1] G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. Henri Poincaré 3 (1967) 433–438.
1
[2] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs–Theory and Application, Academic Press, New York, 1980.
2
[3] A. W. Dudek, On the spectrum of the generalised Petersen graphs, Graphs Combin. 32 (2016) 1843–1850.
3
[4] R. Frucht, J. E. Graver and M. E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211–218.
4
[5] R. Gera and P. Stanica, The spectrum of generalized Petersen graphs, Australas. J. Combin. 49 (2011) 39–45.
5
[6] C. Godsil and G. F. Royle, Algebraic Graph Theory, Springer, New York, 2001.
6
[7] P. D. Powell, Calculating determinants of block matrices, arXiv:1112.4379v1 [math.RA].
7
[8] S. Saadat-Akhtar, S. Sahebi and M. Ghorbani, Co-spectrality of permutation graphs, Ars Combin., in press.
8
[9] V. Yegnanarayanan, On some aspects of the generalized Petersen graph, Electron. J. Graph Theory Appl. 5 (2017) 163–178.
9