On NP-hard graph properties characterized by the spectrum

2014

We will explain basic concepts of spectral graph theory, which studies graph properties via the algebraic characteristics of its adjacency matrix or Laplacian matrix. It turns out that investigating eigenvalues and eigenvectors of a graph provides unexpected and interesting results. Methods of spectral graph theory can be used to examine large random graphs and help tackle many difficult combinatorial problems (in particular because of recently invented algorithms which are able to compute eigenvalues and eigenvectors even of huge graphs in relatively short time). This approach uses highly efficient tools from advanced linear algebra, matrix theory and geometry. This paper will cover the following issues :

The Electronic Journal of Combinatorics

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the 2-clique extension of the $(t+1)\times (t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough.

Linear Algebra and its Applications, 2015

Let G be a graph with n vertices and λ n (G) be the least eigenvalue of its adjacency matrix of G. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.

This paper deals with graphs that are known as multicone graphs. A multicone graph is a graph obtained from the join of a clique and a regular graph. Let w, l, m be natural numbers and k is a natural number. It is proved that any connected graph cospectral with multicone graph Kw mECP k l is determined by its adjacency spectra as well as its Laplacian spectra, where ECP k l = K 3 k , 3 k , ..., 3 k l times. Also, we show that complements of some of these mul-ticone graphs are determined by their adjacency spectra. Moreover, we prove that any connected graph cospectral with these multicone graphs must be perfect. Finally , we pose two problems for further researches. Mathematics Subject Classification (2010): 05C50.

Lecture Notes in Mathematics, 1976

The polynomial of a graph i s the characteristic polynomial of i t s 0-1 acLjacency matrix. Two graphs are c o s p e c t r a l i f their polynomials are the same.

Linear Algebra and its Applications, 2007

Graphs with (k, τ)-regular sets and equitable partitions are examples of graphs with regularity constraints. A (k, τ)-regular set of a graph G is a subset of vertices S ⊆ V (G) inducing a k-regular subgraph and such that each vertex not in S has τ neighbors in S. The existence of such structures in a graph provides some information about the eigenvalues and eigenvectors of its adjacency matrix. For example, if a graph G has a (k 1 , τ 1)-regular set S 1 and a (k 2 , τ 2)-regular set S 2 such that k 1 − τ 1 = k 2 − τ 2 = λ, then λ is an eigenvalue of G with a certain eigenvector. Additionally, considering primitive strongly regular graphs, a necessary and sufficient condition for a particular subset of vertices to be (k, τ)-regular is introduced. Another example comes from the existence of an equitable partition in a graph. If a graph G, has an equitable partition π then its line graph, L(G), also has an equitable partition,π, induced by π, and the adjacency matrix of the quotient graph L(G)/π is obtained from the adjacency matrix of G/π .

arXiv (Cornell University), 2022

We explore several properties of the adjacency matrix of a class of connected cographs. Those cographs can be defined by a finite sequence of natural numbers. Using that sequence we obtain multiplicity of the eigenvalues 0, −1, inertia for the adjacency matrix of the cograph under consideration. We find an eigenvalue-free interval for such graphs which is better than previously obtained interval (−1, 0). Finally, we establish a suitable formula for the characteristic polynomial of the adjacency matrix for such graphs.

Annals of Mathematical Sciences and Applications, 2017

It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.

Linear Algebra and its Applications, 2021

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.