Let G be a graph with adjacency matrix A(G) and let D(G) the diagonal of degrees G. For every real α ∈ [0, 1], Nikiforov [21] Wang et al. [26] defined matrices Aα(G) Lα(G), respectively, as = αD(G)+(1−α)A(G) Lα(G) αD(G)+(α − 1)A(G). In this paper, we obtain some relationships between eigenvalues these for families graphs, part Aα Lα-spectrum spider display Lα-characteristic polynomials when the...

An expansion graph of a directed weighted graph G0 is obtained fromG0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum u...

An expansion graph of a directed weighted graph G0 is obtained fromG0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum u...

In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. ...

We express the discrete Ricci curvature of a graph as minimal eigenvalue family matrices, one for each vertex whose entries depend on local adjacency structure graph. Using this method we compute or bound Cayley graphs finite Coxeter groups and affine Weyl groups. As an application obtain isoperimetric inequality that holds all

Suppose k is a positive integer, G and H are graphs, and / is a k-to-l correspondence from a vertex set of G onto a vertex set of H. Conditions on the adjacency matrices are given that are necessary and sufficient for / to extend to a continuous k-to-l map from G onto H .

We construct self-orthogonal codes from the row span over F2 or F3 of the adjacency matrices of some strongly regular graphs defined by the rank-3 action of the simple unitary group U4(3) on the conjugacy classes of some of its maximal subgroups. We establish some properties of these codes and the nature of some classes of codewords.

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.

In their 1978 paper \Distance Matrix Polynomials of Trees", [4], Graham and Lov asz proved that the coeÆcients of the characteristic polynomial of the distance matrix of a tree (CPD(T )) can be expressed in terms of the numbers of certain subforests of the tree. This result was generalized to trees with weighted edges by Collins, [1], in 1986. Graham and Lov asz computed these coeÆcients for al...

For almost all graphs the answer to the question in the title is still unknown. Here we survey the cases for which the answer is known. Not only the adjacency matrix, but also other types of matrices, such as the Laplacian matrix, are considered. © 2003 Elsevier Inc. All rights reserved.