Survey of Graph Energies
Ivan
Gutman
University Kragujevac, Serbia
author
Boris
Furtula
State University of Novi Pazar, Novi Pazar; Serbia
author
text
article
2017
eng
Let 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
85
129
http://mir.kashanu.ac.ir/article_46658_d162c1bd23ebd5a2f91d1be3caf51c63.pdf
dx.doi.org/10.22052/mir.2017.81507.1057
On Eccentricity Version of Laplacian Energy of a Graph
Nilanjan
De
Calcutta Institute of Engineering and Management
author
text
article
2017
eng
The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. 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 G.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
131
139
http://mir.kashanu.ac.ir/article_46665_5d4ef03c2a66934ca736a18abb23be5f.pdf
dx.doi.org/10.22052/mir.2017.70534.1051
On Relation between the Kirchhoff Index and Laplacian-Energy-Like Invariant of Graphs
Emina
Milovanovic
Faculty of Electronic Engineering, University of Nis, Nis, Serbia
author
Igor
Milovanovic
Faculty of Electronic Engineering, University of Nis, Nis, Serbia
author
Marjan
Matejic
Faculty of Electronic Engineering, University of Nis, Nis, Serbia
author
text
article
2017
eng
Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ1 ≥ μ2 ≥...≥μn-1 >μn=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣi=1n-11/μi and LEL(G)=Σi=1n-1 √μi, respectively. In this paper we consider relationship between Kf(G) and LEL(G).
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
141
154
http://mir.kashanu.ac.ir/article_46678_73d7cbf6e273d4d973106287024507a7.pdf
dx.doi.org/10.22052/mir.2017.85687.1063
The Signless Laplacian Estrada Index of Unicyclic Graphs
Hamid Reza
Ellahi
Department of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R Iran
author
Ramin
Nasiri
Department of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R Iran
author
Gholam Hossein
Fath-Tabar
Department of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Kahsan, Kashan, Iran
author
Ahmad
Gholami
Department of Mathematics,
Faculty of Science,
University of Qom,
Qom, I R Iran
author
text
article
2017
eng
For a simple graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$, where $q^{}_1, q^{}_2, \dots, q^{}_n$ 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
155
167
http://mir.kashanu.ac.ir/article_46679_9ebe519f917a721f6f9fd832c72a83d7.pdf
dx.doi.org/10.22052/mir.2017.57775.1038
More Equienergetic Signed Graphs
Harishchandra S.
Ramane
Department of Mathematics, Karnatak University, Dharwad - 580003, India
author
Mahadevappa M.
Gundloor
Department of Mathematics, Karnatak University, Dharwad - 580003, India
author
text
article
2017
eng
The 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 $n geq 8$ vertices.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
169
179
http://mir.kashanu.ac.ir/article_49308_3be7bf0a0c2de6562035165081ffbebc.pdf
dx.doi.org/10.22052/mir.2017.90820.1068
Seidel Signless Laplacian Energy of Graphs
Harishchandra
Ramane
Karnatak University
author
Ivan
Gutman
University Kragujevac, Serbia
author
Jayashri
Patil
Hirasugar Institute of Technology
author
Raju
Jummannaver
Karnatak University
author
text
article
2017
eng
Let $S(G)$ be the Seidel matrix of a graph $G$ of order $n$ and let $D_S(G)=diag(n-1-2d_1, n-1-2d_2,\ldots, n-1-2d_n)$ 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 $SL^+(G)=D_S(G)+S(G)$. The Seidel signless Laplacian energy $E_{SL^+}(G)$ is defined as the sum of the absolute deviations of the eigenvalues of $SL^+(G)$ from their mean. In this paper, we establish the main properties of the eigenvalues of $SL^+(G)$ and of $E_{SL^+}(G)$.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
181
191
http://mir.kashanu.ac.ir/article_53998_01ab0ae77936bf1f5161db2349204526.pdf
dx.doi.org/10.22052/mir.2017.101641.1081
Eigenvalues of the Cayley Graph of Some Groups with respect to a Normal Subset
Maryam
Jalali-Rad
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran
author
text
article
2017
eng
Set 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 V8n = < a, b | a^{2n} = b^{4} = e, aba = b^{-1}, ab^{-1}a = b>,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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
193
207
http://mir.kashanu.ac.ir/article_53999_50a22096d7b267c4ef41bd53e9f89c1e.pdf
dx.doi.org/10.22052/mir.2017.101675.1082
Laplacian Sum-Eccentricity Energy of a Graph
Biligirirangaiah
Sharada
Mysore University, Mysore, India
author
Mohammad Issa
Sowaity
Mysore University, Mysore, India
author
Ivan
Gutman
University Kragujevac, Serbia
author
text
article
2017
eng
We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=\sum_{i=1}^n |\eta_i|, where \eta_i=\zeta_i-\frac{2m}{n} and where \zeta_1,\zeta_2,\ldots,\zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if \sum_{i=1}^n |\eta_i|=\sum_{i=1}^n |\zeta_i|. Conditions for the existence of such graphs are established.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
2
v.
2
no.
2017
209
219
http://mir.kashanu.ac.ir/article_54000_a60755aff16ea35f3f068051c0878426.pdf
dx.doi.org/10.22052/mir.2017.106176.1084