The Q-generating Function for Graphs with Application

Abstract

For a simple connected graph G, the Q-generating function of the numbers $N_k$ of semi-edge walks of length k in G is defined by $W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$. This paper reveals that the Q-generating function $W_Q(t)$ may be expressed in terms of the Q-polynomials of the graph G and its complement $\overline{G}$. Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $W_Q(t)$, we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $(\lambda I_n-Q(G))^{-1}$. This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.