Abstract
Let \(G=(U,V)\) be a connected bipartite graph and let \(C(G)\) be the algebraic structure count of \(G\). Gutman's formulas in [12] states that for any edge \(ab\) of \(G\), then there exists an \(\varepsilon\in \{1,-1\}\) such that
\[
C(G)=|c(G-ab)+\varepsilon C(G-a-b)|.
\]
The current author extended the above result and obtained some variants of Gutman's formulas in [21,22] as follows.
1. For any \(a,c\in U,b,d\in V\), then there exists an \(\varepsilon_1\in \{1,-1\}\) such that
\[
C(G)C(G-a-b-c-d)=|C(G-a-b)C(G-c-d)+\varepsilon_1 C(G-a-d)C(G-b-c)|.
\]
2. For any 2-matching \(\{u_1v_1,u_2v_2\}\) of \(G\), then there exists an \(\varepsilon_2 \in \{1,-1\}\) such that
\[
C(G)C(G-u_1v_1-u_2v_2)=|C(G-u_1v_1)C(G-u_2v_2)+\varepsilon_2
C(G-u_1-v_2)C(G-u_2-v_1)|,
\]
where \(u_1,u_2\in U, v_1,v_2\in V\).
3. For any edge \(yz\) and two vertices \(r\) and \(s\) of \(G\) satisfying \(y,r\in U\) and \(z,s\in V\) and \(\{y,z\}\cap \{r,s\}=\emptyset\), then there exists an \(\varepsilon_3\in \{1,-1\}\) such that
\[
C(G)C(G-yz-r-s)=|C(G-yz)C(G-r-s)+\varepsilon_3 C(G-y-s)C(G-r-z)|.
\]
In this note, we prove that, if \(|U|=|V|=n\), then there exists a \(\beta=(\nu_1,\nu_2,\ldots,\nu_{m})\) satisfying \(\nu_1,\nu_2,\ldots,\nu_{m}\in \{1,-1\}\) such that
\[
(m-n)C(G)=\left|\sum_{i=1}^m\nu_iC(G-e_i)\right|,
\]
where the sum ranges over all edges \(e_1,e_2,\ldots,e_m\) of \(G\).
