It is well known that spectral Turán type problem one of the most classical problems in graph theory. In this paper, we consider problem. Let G be a and let set graphs, say G-free if does not contain any element as subgraph. Denote by ?1 ?2 largest second eigenvalues adjacency matrix A(G) G, respectively. paper focus on characterization graphs without short odd cycles according to graphs. First...

We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint un...

Suppose that is a graph with vertices , edges free group action on the finitely many orbits, and linear operator of finite order Hilbert space such commutes action. Fix in pure-point spectrum consider vector all eigenfunctions support . Then nontrivial generated module over ring Laurent polynomials. We show density given by an Euler-characteristic-type formula taking resolution Furthermore, we ...

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}. ...