Symmetry and two symmetry measures for the web and spider web graphs

Abstract

Suppose G is a simple graph, $\Gamma \le Aut(G)$ and $\alpha \in \Gamma$. Define $\mu (G)=\sum _{u \in V(G), \alpha \in \Gamma } d(u,\alpha (u))$, $\eta (G)=\sum _{u \in V(G), \alpha \in \Gamma } ( d(u,\alpha (u)))^2$, $GP(G)= \frac{|V(G)|}{2|\Gamma |}\mu (G)$ and $GP^{(2)}(G) = \frac{1}{2}GP(G) + \frac{|V(G)|}{4|\Gamma |}\eta (G)$. The graph invariants GP(G) and $GP^{(2)}(G)$ are two recent measures for comparing symmetry of complex networks. The aim of this paper is to compute the symmetry of web and spider web graphs and then apply the structure of these groups to calculate the graph invariants GP(G) and $GP^{(2)}(G)$. These numbers help us to judge on the complexity of these networks.