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.

对于一个简单的连通图G ^，所述Q的数字-产生功能\（N_k \）半边的长度的走ķ在ģ由下式定义\（W_Q（T）= \和\ nolimits _ {K = 0} ^ \ infty {N_k t ^ k} \）。本文揭示了Q生成函数\（W_Q（t）\）可以用图G及其补\（\ overline {G} \）的Q多项式表示。利用这个结果，我们研究图的一些Q谱性质并计算Q-从各种图形操作获得的某些图形的多项式，例如正则图的补图，两个图的连接和两个图的（边）电晕。作为Q生成函数\（W_Q（t）\）的另一个应用，我们还给出了G的Q冠冕的组合解释，其定义为矩阵\（（\ lambda I_n-Q（G））^ {-1} \）。该结果可用于获得两个图的（边缘）电晕的Q多项式的许多替代计算。此外，我们还计算了两个图和完整的多部分图的连接的Q生成函数。