Enumeration of Switching Non-isomorphic Signed Wheels

Abstract

Two signed graphs are called switching isomorphic to each other if one is isomorphic to a switching of the other. The wheel $W_n$ is the join of the cycle $C_n$ and a vertex. For $0\le p\le n$, ${\psi }_p(n)$ is defined to be the number of switching non-isomorphic signed $W_n$ with exactly $p$ negative edges on $C_n$. The number of switching non-isomorphic signed $W_n$ is denoted by $\psi (n)$. In this paper, we compute the values of ${\psi }_p(n)$ for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,n$ and of $\psi (n)$ for $n=4,5,...,10$. Our method of obtaining ${\psi }_p(n)$ not only count the switching non-isomorphic signed wheels but also generates them.