On the automorphism group of an infinite graph.
On the bounds of Laplacian eigenvalues of -connected graphs
Let be the algebraic connectivity, and let be the Laplacian spectral radius of a -connected graph with vertices and edges. In this paper, we prove that with equality if and only if is the complete graph or . Moreover, if is non-regular, then where stands for the maximum degree of . Remark that in some cases, these two inequalities improve some previously known results.
On the characterization of graphs with pendent vertices and given nullity.
On the common-neighborhood energy of a graph
On the complexity of the subgraph problem
On The Determinant of q-Distance Matrix of a Graph
In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph...
On the distance polynomial of a graph
Several properties of the distance polynomial are discussed.
On the distance spectrum of a cycle
Analytic expressions for the roots of the distance polynomial of a cycle are given.
On the energy of unitary Cayley graphs.
On the estrada index of graphs with given number of cut edges.
On the exponent of -regular primitive matrices.
On the first eigenvalue of bipartite graphs.
On the functions with values in .
On the inverse eigenvalue problem for a special kind of acyclic matrices
We study an inverse eigenvalue problem (IEP) of reconstructing a special kind of symmetric acyclic matrices whose graph is a generalized star graph. The problem involves the reconstruction of a matrix by the minimum and maximum eigenvalues of each of its leading principal submatrices. To solve the problem, we use the recurrence relation of characteristic polynomials among leading principal minors. The necessary and sufficient conditions for the solvability of the problem are derived. Finally, a...
On the inverse eigenvalue problems: the case of superstars.
On the inverse of the adjacency matrix of a graph
A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary...
On the kernel of tree incidence matrices.
On the Laplacian energy of a graph
In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on vertices . Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant , and completely describe this class in the case .
On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph
The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively....
On the Laplacian spread of trees and unicyclic graphs