A Class of L-Cospectral Graphs are Isomorphic

Abstract

The spectral graph theory is a theory in which graphs are studied by means of their adjacency eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of associated graph matrix. What kind of graphs are determined by their spectra is a difficult problem in graph theory. Let 1 2 3 { , ( , , )} n q H C T n n n be a graph of order n obtained by identifying a vertex of cycle q C with the vertex of degree three of T-shape three 1 2 3 ( , , ) T n n n . In this paper, we will prove that for q even, all the L-cospectral graphs 1 2 3 { , ( , , )} n q H C T n n n are isomorphic.