The Q-generating Function for Graphs with Application

For a simple connected graph G, the Q-generating function of the numbers Nkof semi-edge walks of length kin Gis defined by WQ(t)=∑k=0∞Nktk. This paper reveals that the Q-generating function WQ(t)may be expressed in terms of the Q-polynomials of the graph Gand its complement 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 WQ(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 (λIn-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.