by $G_{(V,E)}$, or just $G_E$ if $V$ consists of the end vertices of edges in $E$. The same notation is used when $E = (e_1,e_2, \ldots ,e_k)$ is a tuple of edges, some of which may be identical. Isomorphism of two graphs $G$ and $H$ is denoted by $G\cong H$. For $i > 0$, a graph isomorphic to a cycle of length $i$ is denoted by $C_i$, and the number of cycles of length $i$ in $G$ is denoted by $\psi_i(G)$, where, as a convention, $C_i \cong K_i$ for $i \in \{1,2\}$. The number of hamiltonian cycles is denoted by a special symbol $ham(G)$ instead of $\psi_{v(G)}(G)$. While counting the number of subgraphs of a graph $G$ that are isomorphic to a graph $F$, it is important to make a distinction between induced subgraphs and edge subgraphs. The number of subgraphs of $G$ that are isomorphic to $F$ is denoted by ${\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}}$, and the number of induced subgraphs of $G$ that are isomorphic to $F$ is denoted by $\displaystyle\binom{G}{F}$. The two numbers are related by $${\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}} = \sum_{H|VH=VF}\displaystyle\binom{G}{H}{\mbox{$\left[\begin{array}{c}H\\#2\end{array}\right]$}}$$ where the summation is over distinct isomorphism types of graphs $H$. The characteristic polynomial of $G$ is denoted by $P(G;\lambda) = \sum_{i 