Graphs whose Spectrum Determined by Non-constant Coefficients

Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M ( G , x ) = ∑ i = 0 n α i ( G ) x n − i . We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with a i ( H ) = a i ( G ) , 0 ⩽ i ⩽ n − 1 , then S p e c ( G ) = S p e c ( H ) (if M is the adjacency matrix or the Laplacian matrix of G, then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we show that there are infinitely many pairs ( T , T ′ ) of trees such that A ( T , x ) − A ( T ′ , x ) = − 1 .