Abstract
Suppose that \( G \) is a connected bipartite graph with bipartition \( (U,V) \) and \( f(G) \) be the algebraic structure count of \( G \). Gutman [Note on algebraic structure count, Z. Naturforsch.39a (1984) 794--796.]
proved that, if \( uv \) is an edge of \( G \), then there exists an \( \epsilon\in \{1,-1\} \) such that \begin{equation} f(G)=|f(G-uv)+\epsilon f(G-u-v)|. \tag{i} \end{equation} Ye [Further variants of Gutman's formulas, MATCH Commun.
Math. Comput. Chem.90 (2023) 235--245.] obtained a variant of the Gutman's formula above and proved 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 \[ (m-n)f(G)=\left|\sum_{i=1}^m\nu_if(G-e_i)\right|, \tag{ii} \] where the sum ranges over all edges \( e_1,e_2,\ldots,e_m \) of \( G \). Both formulae \( (i) \) and \( (ii) \) are linear recurrences. But it is difficult
to determine \( \epsilon=1 \) or \( -1 \) in \( (i) \) and \( \nu_i=1 \) or \( -1 \) in \( (ii) \). In this paper, we obtain a quadratic recurrence of the algebraic structure count of \( G \) as follows. \[ (|E(G)|-2n)f^2(G)=\sum_{uv\in
E(G)}[f^2(G-uv)-f^2(G-u-v)], \tag{iii} \] where the sum ranges over all edges of \( G \). Meanwhile, we obtain a quadratic recurrence of the number of perfect matchings of \( G \) which is similar to formula \( (iii) \).
