Abstract
Let \(C(G)\) denote the algebraic structure count of a bipartite graph \(G\) with bipartition \((V_1,V_2)\). Gutman proved that, for any edge \(e=ab\) of \(G\), one of the following formulas holds:
\[
\begin{aligned}
C(G)&=C(G-e)+C(G-a-b),\\
C(G)&=C(G-e)-C(G-a-b),\\
C(G)&=C(G-a-b)-C(G-e).
\end{aligned}
\]
In this paper, we prove that, for any pair of independent edges \(\{f=uv,g=wx\}\) of \(G\), then one of the following formulas holds.
\[
\begin{aligned}
C(G)C(G-f-g)&=C(G-f)C(G-g)+C(G-u-x)C(G-w-v),\\
C(G)C(G-f-g)&=C(G-f)C(G-g)-C(G-u-x)C(G-w-v),\\
C(G)C(G-f-g)&=C(G-u-x)C(G-w-v)-C(G-f)C(G-g),
\end{aligned}
\]
where \(u,w\in V_1, v,x\in V_2\). We prove also that, for any edge \(h=yz\) and two vertices \(r\) and \(s\) such that \(y,r\in V_1\) and \(z,s\in V_2\) and \(\{y,z\}\cap \{r,s\}=\emptyset\), then one of the following formulas holds.
\[
\begin{aligned}
C(G)C(G-h-r-s)&=C(G-h)C(G-r-s)+C(G-y-s)C(G-r-z),\\
C(G)C(G-h-r-s)&=C(G-h)C(G-r-s)-C(G-y-s)C(G-r-z),\\
C(G)C(G-h-r-s)&=C(G-y-s)C(G-r-z)-C(G-h)C(G-r-s)$.
\end{aligned}
\]
