Abstract
The algebraic structure count of a bipartite graph \(G = (U, V)\), denoted by \(L(G)\), is defined as the difference between the number of so-called "even" and "odd" Kekulé structures of G by Wilcox in theoretical organic chemistry. Let \(e = uv\) be an edge of a bipartite graph \(G\). Gutman proved that G satisfies one of the following relations:
\[
\begin{align}
L(G) &= L(G − e) + L(G − u − v), \\
L(G) &= L(G − e) − L(G − u − v), \\
L(G) &= −L(G − e) + L(G − u − v),
\end{align}
\]
where \(G − e\) (resp. \(G − u − v\)) is the graph obtained from \(G\) by deleting edge \(e\) (resp. vertices \(u\) and \(v\)). In this short note, we obtain a similar result and prove that for any \(u_1, u_2 \in U, v_1, v_2 \in V\), \(G\) satisfies one of the following relations:
\[
\begin{align}
L(G)L(G − u_1 − u_2 − v_1 − v_2) &= L(G − u_1 − v_1)L(G − u_2 − v_2) + L(G − u_1 − v_2)L(G − u_2 − v_1), \\
L(G)L(G − u_1 − u_2 − v_1 − v_2) &= L(G − u_1 − v_1)L(G − u_2 − v_2) − L(G − u_1 − v_2)L(G − u_2 − v_1), \\
L(G)L(G − u_1 − u_2 − v_1 − v_2) &= −L(G − u_1 − v_1)L(G − u_2 − v_2) + L(G − u_1 − v_2)L(G − u_2 − v_1).
\end{align}
\]
