Graph Isomorphism is a widely studied problem due to its practical applications in various fields of networks, chemistry and finger print detection, recent problems in biology such as diabetes detection, protein structure and information retrieval. An approach to the graph isomorphism detection is based on vertex invariant. In the existing approach vertex invariants is used to partition the mat...

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Spectral graph theory seeks to relate the eigenvectors and eigenvalues of matrices corresponding to a Graph to the combinatorial properties of the graph. While generally the theorems are designed for unweighted and undirected graphs they can be extended to the weighted graph case and (less co...

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, 2 , of (the adjacency matrix of) a random d-regular graph, G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at the k-th step, for various values of k. Our main theorem about eigenvalues is that E fj 2 (G)j m g 2 p 2d ? ...

For a finite connected simple graph, the Terwilliger algebra is matrix generated by adjacency and idempotents corresponding to distance partition with respect fixed vertex. We will consider algebras defined two other partitions centralizer of stabilizer vertex in automorphism group graph. give some methods compute such examples for various graphs.

The rank of the adjacency matrix of a graph over the real field is a natural parameter associated with the graph, and there has been substantial interest in relating this to other graph theoretic parameters. In particular the relationship between the rank and chromatic number of a graph, and the rank and the maximum number of vertices of a graph has been extensively studied. This was at least i...

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we prov...

We initiate the study of pretty good quantum fractional revival in graphs, a generalization state transfer graphs. give complete characterization graph terms eigenvalues and eigenvectors adjacency matrix graph. This follows from lemma due to Kronecker on Diophantine approximation, is similar spectral Using this, we characterizations when can occur paths cycles.

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ǫ are called graphs with defect or excess ǫ, respectively. While Moore graphs (graphs with ǫ = 0) and graphs with defect or excess 1 have been characte...

Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of...

Hoffman proved that for a simple graph G, the chromatic number χ(G) obeys χ(G) ≥ 1 − λ1 λn where λ1 and λn are the maximal and minimal eigenvalues of the adjacency matrix of G respectively. Lovász later showed that χ(G) ≥ 1− λ1 λn for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lovász’s theorem, then extend the technique to derive generali...