University of KashanMathematics Interdisciplinary Research2538-36392220171201Survey of Graph Energies851294665810.22052/mir.2017.81507.1057ENIvanGutmanUniversity Kragujevac, SerbiaBorisFurtulaState University of Novi Pazar, Novi Pazar, SerbiaJournal Article20160314Let graph energy is a graph--spectrum--based quantity, introduced in the 1970s. After a latent period of 20--30 years, it became a popular topic of research both in mathematical chemistry and in ``pure'' spectral graph theory, resulting in over 600 published papers. Eventually, scores of different graph energies have been conceived. In this article we provide the basic facts on graph energies, in particular historical and bibliographic data.University of KashanMathematics Interdisciplinary Research2538-36392220171201On Eccentricity Version of Laplacian Energy of a Graph1311394666510.22052/mir.2017.70534.1051ENNilanjanDeDepartment of Basic Sciences and Humanities (Mathematics),
Calcutta Institute of Engineering and Management,
Kolkata, West Bengal,
IndiaJournal Article20161217The energy of a graph <em>G</em> is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of <em>G</em>, whereas the Laplacian energy of a graph <em>G</em> is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of <em>G</em> and the average degree of the vertices of <em>G</em>. Motivated by the work from Sharafdini and Panahbar [R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and other topological indices. J. Math. Nanosci. 2016, 6, 57-65], in this paper we investigate the eccentricity version of Laplacian energy of a graph <em>G</em>.University of KashanMathematics Interdisciplinary Research2538-36392220171201On Relation between the Kirchhoff Index and Laplacian-Energy-Like Invariant of Graphs1411544667810.22052/mir.2017.85687.1063ENEminaMilovanovicFaculty of Electronic Engineering, University of Nis, Nis, Serbia0000-0002-1905-4813IgorMilovanovicFaculty of Electronic Engineering, University of Nis, Nis, Serbia0000-0003-2209-9606MarjanMatejicFaculty of Electronic Engineering, University of Nis, Nis, Serbia0000-0003-0354-6749Journal Article20170511Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ<sub>1</sub> ≥ μ<sub><span style="font-size: 8.33333px;">2 </span></sub>≥...≥μ<sub><span style="font-size: 8.33333px;">n-1 </span></sub>>μ<sub>n</sub>=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣ<sub>i=1</sub><sup>n-1</sup>1/μ<sub><span style="font-size: 8.33333px;">i </span></sub>and LEL(G)=Σ<sub>i=1</sub><sup>n-1</sup> √μ<sub>i</sub>, respectively. In this paper we consider relationship between Kf(G) and LEL(G).University of KashanMathematics Interdisciplinary Research2538-36392220171201The Signless Laplacian Estrada Index of Unicyclic Graphs1551674667910.22052/mir.2017.57775.1038ENHamid RezaEllahiDepartment of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R IranRaminNasiriDepartment of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R IranGholam HosseinFath-TabarDepartment of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Kahsan, Kashan, IranAhmadGholamiDepartment of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R IranJournal Article20160714For a simple graph G, the signless Laplacian Estrada index is defined as SLEE(G)=∑<sup>n</sup><sub>i=1</sub>e<sup>qi</sup>, where q<sub>1</sub>, q<sub>2</sub>,..., q<sub>n</sub> are the eigenvalues of the signless Laplacian matrix of G. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest SLEE's and then determine the unique unicyclic graph with maximum SLEE among all unicyclic graphs on n vertices with a given diameter. All extremal graphs, which have been introduced in our results are also extremal with respect to the signless Laplacian resolvent energy.University of KashanMathematics Interdisciplinary Research2538-36392220171201More Equienergetic Signed Graphs1691794930810.22052/mir.2017.90820.1068ENHarishchandra S.RamaneDepartment of Mathematics, Karnatak University, Dharwad - 580003, IndiaMahadevappa M.GundloorDepartment of Mathematics, Karnatak University, Dharwad - 580003, IndiaJournal Article20170629The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction of unbalanced, noncospectral equieneregtic signed graphs on n ≥ 8 vertices.University of KashanMathematics Interdisciplinary Research2538-36392220171201Seidel Signless Laplacian Energy of Graphs1811915399810.22052/mir.2017.101641.1081ENHarishchandra S.RamaneDepartment of Mathematics,
Karnatak University,
Dharwad – 580003, IndiaIvanGutmanFaculty of Science,
University of Kragujevac,
P.O. Box 60, 34000 Kragujevac, SerbiaJayashri B.PatilDepartment of Mathematics,
Hirasugar Institute of Technology,
Nidasoshi – 591236, IndiaRaju B.JummannaverDepartment of Mathematics,
Karnatak University,
Dharwad – 580003, IndiaJournal Article20171020Let S(G) be the Seidel matrix of a graph G of order n and let D<sub>S</sub>(G)=diag(n-1-2d<sub>1</sub>, n-1-2d<sub>2</sub>,..., n-1-2d<sub>n</sub>) be the diagonal matrix with d_i denoting the degree of a vertex v_i in G. The Seidel Laplacian matrix of G is defined as SL(G)=D_S(G)-S(G) and the Seidel signless Laplacian matrix as SL<sup>+</sup>(G)=D<sub>S</sub>(G)+S(G). The Seidel signless Laplacian energy E<sub>SL+</sub>(G) is defined as the sum of the absolute deviations of the eigenvalues of SL<sup>+</sup>(G) from their mean. In this paper, we establish the main properties of the eigenvalues of SL<sup>+</sup>(G) and of E<sub>SL+</sub>(G).University of KashanMathematics Interdisciplinary Research2538-36392220171201Eigenvalues of the Cayley Graph of Some Groups with respect to a Normal Subset1932075399910.22052/mir.2017.101675.1082ENMaryamJalali-RadDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, IranJournal Article20170821Set X = { M11, M12, M22, M23, M24, Zn, T4n, SD8n, Sz(q), G2(q), V8n}, where M11, M12, M22, M23, M24 are Mathieu groups and Zn, T4n, SD8n, Sz(q), G2(q) and V8n denote the cyclic, dicyclic, semi-dihedral, Suzuki, Ree and a group of order 8n presented by <br /> V8n = < a, b | a<sup>2n</sup> = b<sup>4</sup> = e, aba = b<sup>-1</sup>, ab<sup>-1</sup>a = b>,<br />respectively. In this paper, we compute all eigenvalues of Cay(G,T), where G in X and T is minimal, second minimal, maximal or second maximal normal subset of G{e} with respect to its size. In the case that S is a minimal normal subset of G{e}, the summation of the absolute value of eigenvalues, energy of the Cayley graph, are evaluated.University of KashanMathematics Interdisciplinary Research2538-36392220171201Laplacian Sum-Eccentricity Energy of a Graph2092195400010.22052/mir.2017.106176.1084ENBiligirirangaiahSharadaDepartment of Studies in Mathematics,
University of Mysore, Manasagangotri,
Mysuru – 570 006, IndiaMohammad IssaSowaityDepartment of Studies in Mathematics,
University of Mysore, Manasagangotri,
Mysuru – 570 006, IndiaIvanGutmanFaculty of Science,
University of Kragujevac,
P.O. Box 60, 34000 Kragujevac, SerbiaJournal Article20171119We introduce the Laplacian sum-eccentricity matrix <strong>LS<sub>e</sub></strong> of a graph G, and its Laplacian sum-eccentricity energy LS<sub>e</sub>E=∑<sup>n</sup><sub>i=1</sub>|η<sub>i</sub>|, where η<sub>i</sub>=ξ<sub>i</sub>-(2m/n) and where ξ<sub>1</sub>,ξ<sub>2</sub>,...,ξ<sub>n </sub>are the eigenvalues of <strong>LS<sub>e</sub></strong>. Upper bounds for LS<sub>e</sub>E are obtained. A graph is said to be twinenergetic if ∑<sup>n</sup><sub>i=1</sub>|η<sub>i</sub>|=∑<sup>n</sup><sub>i=1</sub>|ξ<sub>i</sub>|. Conditions for the existence of such graphs are established.