Two Parameters Characteristic Polynomial And Its Applications

Let G be a simple graph with n vertices. We denote by V (G) the vertex set and by E (G) the edge set of G. The adjacency matrix A( G) of G is the n×n matrix with (u ,v)th entry equal to 1 if vertices u and v are adjacent and equal to 0 otherwise. Denote by D (G) the diagonal matrix with degrees of the corresponding vertices of G on the main diagonal. Denote the characteristic polynomial of the adjacency matrix A(G) by PA ( G,λ)= b0λn+b1λn-1 ++bn1λ+bn. The eigenvalues of A( G) and the spectrum of A( G) are also called the adjacency eigenvalues of G and the adjacency spectrum of G .A graph G is said to be determined by its adjacency spectrum if any graph having the same adjacency spectrum as G is isomorphic to G . In this paper, by using the coefficients of characteristic polynomials of the adjacency matrix A( G) we give two invariantsΠ1 (G) andΠ2 (G), whereThen some properties of the invariants are given. And we characterize all connected graphs withΠ1 (G)=1, 0, -1, -2, -3 andΠ2 (G)=0, -1, -2,-3. At last, by using the property of the invariantsΠ1 (G) andΠ2 (G), We proved that some graphs are determined by their adjacency spectrum.