Spectral Properties with the Difference between Topological Indices in Graphs

Let G be a graph of order n with vertices labeled as v1,v2,…,vn. Let di be the degree of the vertex vi, for i=1,2,…,n. The difference adjacency matrix of G is the square matrix of order n whose i,j entry is equal to di+dj−2−1/didj if the vertices vi and vj of G are adjacent or vivj∈EG and zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied. We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph.

The Energy Change of the Complete Multipartite Graph

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. [S. Akbari, E. Ghorbani, and M. Oboudi. Edge addition, singular values, and energy of graphs and matrices. {\em Linear Algebra Appl.}, 430:2192--2199, 2009.] proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, a new method to study the energy of graph is explored. As an application of this new method, the above natural question is answered and it is completely determined how the energy of a complete multipartite graph changes when one edge is removed.

The Energy of Cayley Graphs for Symmetric Groups of Order 24

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

On the Seidel Energy of Certain Mesh Derived Networks

The energy of graph G is defined as the sum of the absolute values of eigenvalues of the adjacency matrix A(G). The manual calculation of energy of graphs consumes several man hours. In this paper, we use MATLAB to generate the Seidel matrix and hence calculate the Seidel energy of some mesh derived networks.

Face Recognition Through Symbolic Modeling of Face Graphs and Texture

Face recognition helps in authentication of the user using remotely acquired facial information. The dynamic nature of face images like pose, illumination, expression, occlusion, aging, etc. degrades the performance of the face recognition system. In this paper, a new face recognition system using facial images with illumination variation, pose variation and partial occlusion is presented. The facial image is described as a collection of three complete connected graphs and these graphs are represented as symbolic objects. The structural characteristics, i.e. graph spectral properties, energy of graph, are extracted and embedded in a symbolic object. The texture features from the cheeks portions are extracted using center symmetric local binary pattern (CS-LBP) descriptor. The global features of the face image, i.e. length and width, are also extracted. Further symbolic data structure is constructed using the above features, namely, the graph spectral properties, energy of graph, global features and texture features. User authentication is performed using a new symbolic similarity metric. The performance is investigated by conducting the experiments with AR face database and VTU-BEC-DB multimodal database. The experimental results demonstrate an identification rate of 95.97% and 97.20% for the two databases.

A note on the Laplacian resolvent energy of graphs

Let [Formula: see text] be a simple connected graph with [Formula: see text] vertices, [Formula: see text] edges and let [Formula: see text] be its Laplacian eigenvalues. The Laplacian resolvent energy of graph [Formula: see text] is defined by [Formula: see text]. In this paper, we give some new lower bounds for the invariant [Formula: see text].

Bounds on Energy and Laplacian Energy of Graphs

Let G be simple graph with n vertices and m edges. The energy E(G) of G, denotedby E(G), is dened to be the sum of the absolute values of the eigenvalues of G. Inthis paper, we present two new upper bounds for energy of a graph, one in terms ofm,n and another in terms of largest absolute eigenvalue and the smallest absoluteeigenvalue. The paper also contains upper bounds for Laplacian energy of graph.