# Mathematics Interdisciplinary Research Volume:2 Issue: 2, Autumn 2017

• تاریخ انتشار: 1396/09/10
• تعداد عناوین: 8
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• Ivan Gutman *, Boris Furtula Pages 85-129
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.‎
Keywords: Energy, spectrum, Graph
• Nilanjan De * Pages 131-139
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.
Keywords: Eccentricity, Eigenvalue, energy (of graph), Laplacian energy, topological index
• Emina Milovanovic *, Igor Milovanovic, Marjan Matejic Pages 141-154

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).the formula is not displayed correctly!

Keywords: Kirchhoff index, Laplacian-energy-like invariant, Laplacian eigenvalues of graph
• Hamid Reza Ellahi, Ramin Nasiri, GholamHossein Fath Tabar, Ahmad Gholami * Pages 155-167

‎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.the formula is not displayed correctly!

Keywords: ‎Signless Laplacian Estrada index‎, ‎unicyclic graphs‎, ‎extremal graphs‎, ‎diameter, ‎signless Laplacian resolvent energy‎
• Harishchandra S. ‎Ramane *, Mahadevappa M. Gundloor Pages 169-179
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.
Keywords: Signed graph_energy of a graph_equienergetic graphs
• Harishchandra Ramane *, Ivan Gutman, Jayashri Patil, Raju Jummannaver Pages 181-191

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)\$.the formula is not displyed correctly!

Keywords: Seidel Laplacian eigenvalues, Seidel Laplacian energy, Seidel signless Laplacian matrix, Seidel signless Laplacian eigenvalues, Seidel signless Laplacian energy
• Maryam Jalali Rad * Pages 193-207

‎‎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.the formula is not displayed correctly!

Keywords: Simple group‎, ‎Cayley graph‎, ‎eigenvalue‎, ‎energy