We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H-eigenva...

Infrared spectra (IR) were used as regressors for a number of QSARs and compared with both mechanistically oriented descriptors and heuristic "chemically neutral" descriptors (modified adjacency matrices eigenvalues). IR spectra usually gave results inferior to those obtained with the mechanistically driven descriptors, with one notable exception, and comparable to those obtained by adjacency m...

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d+1 distinct eigenvalues.

In this paper, we propose a heuristic for the graph isomorphism problem that is based on the eigendecomposition of the adjacency matrices. It is well known, that the eigenvalues of the adjacency matrices of isomorphic graphs need to be identical. However, two graphs GA and GB can be isospectral but non-isomorphic. If the graphs possess repeated eigenvalues, which typically correspond to graph s...

A mixed graph is called second kind hermitian integral (or HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. Eisenstein (0, 1)-adjacency Let ? be an abelian group. We characterize set S for which a Cayley Cay(?,S) HS-integral. also show that and only it

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

Our goal is to use the properties of the adjacency/Laplacian matrix of graphs to first understand the structure of the graph and, based on these insights, to design efficient algorithms. The study of algebraic properties of graphs is called algebraic graph theory. One of the most useful algebraic properties of graphs are the eigenvalues (and eigenvectors) of the adjacency/Laplacian matrix.

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show it is only C6. also there are three kinds of the second largest five eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.

Let n be any positive integer and Fn be the friendship (or Dutch windmill) graph with 2n+1 vertices and 3n edges. Here we study graphs with the same adjacency spectrum as Fn. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let G be a graph cospectral with Fn. Here we prove that if G has no cycle of length 4 or 5, then G ∼= Fn. Moreover if G...

In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues. © 2003 Elsevier Ltd. All rights reser...