We give several old and some new applications of eigenvalue interlacing to matrices associated to graphs. Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (colclique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the Standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity resu...

• Let G be a graph. For subset X of V ( ), the switching σ is signed graph obtained from by reversing signs all edges between and ) ∖ . A( adjacency matrix An eigenvalue called main if it has an eigenvector sum whose entries not equal to zero. Two equivalent graphs share same spectrum, while they may have different eigenvalues. Akbari et al. (2021) conjectured that let ≠ K 2 , 4 { e } Then ther...

For ??[0,1], let A?(G?)=?D(G)+(1??)A(G?), where G is a simple undirected graph, D(G) the diagonal matrix of its vertex degrees and A(G?) adjacency signed graph G? whose underlying G. In this paper, basic properties A?(G?) are obtained, positive semidefiniteness studied some bounds on eigenvalues derived—in particular, lower upper largest eigenvalue obtained.

The energy of a graph G, denoted by E(G), is the sum of the absolute values of all eigenvalues of G . In this paper we present some lower and upper bounds for E(G) in terms of number of vertices, number of edges, and determinant of the adjacency matrix. Our lower bound is better than the classical McClelland’s lower bound. In addition, Nordhaus–Gaddum type results for E(G) are established.

In this paper, all connected bipartite graphs are characterized whose third largest Laplacian eigenvalue is less than three. Moreover, the result is used to characterize all connected bipartite graphs with exactly two Laplacian eigenvalues not less than three, and all connected line graphs of bipartite graphs with the third eigenvalue of their adjacency matrices less than one. c © 2003 Elsevier...

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

The A α matrix of a graph G is defined by ( ) = D + 1 − , 0 ≤ where the diagonal degrees and adjacency . -spectrum denoted S p e c set eigenvalues together with their multiplicities said to be determined generalized for short), if any H ¯ isomorphic In this paper, we present simple arithmetic condition an almost -controllable being which generalizes main results in [17]

The sum of the absolute values eigenvalues graph’s adjacency matrix is known as its ordinary energy. Based on a range other graph matrices, several equivalent energies are being considered. In this work, we considered energy, Laplacian, Randi´c, incidence, and Sombor energy to analyze their relationship using polynomial regression. performance each model exceptional with cross-validation RMSE m...

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

Several inequalities on vertex degrees, eigenvalues, Laplacian eigen-values, and signless Laplacian eigenvalues of graphs are presented in this note. Some of them are generalizations of the inequalities in [2]. We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [1]. We use [n] to denote the set of { 1, 2, ..., n}. ...