ON Q–INTEGRAL (3, s)–SEMIREGULAR BIPARTITE GRAPHS

Let G be a simple graph with adjacency matrix A (= AG). The eigenvalues and the spectrum of A are also called the eigenvalues and the spectrum of G, respectively. If we consider a matrix Q = D + A instead of A, where D is the diagonal matrix of vertex–degrees (in G), we get the signless Laplacian eigenvalues and the signless Laplacian spectrum, respectively. For short, the signless Laplacian eigenvalues and the signless Laplacian spectrum will be called the Q–eigenvalues and the Q–spectrum, respectively. We say that a graph is Q–integral if its signless Laplacian spectrum consists entirely of integers. Let R (= RG) be the n×m vertex–edge incidence matrix of G. Denote by L(G) the line graph of G (recall, vertices of L(G) are in one–to–one correspondence with edges of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G are adjacent). The following relations are well known (see, for example, [2]): RR = AG + D, R R = AL(G) + 2I.