On 4-Sachs Optimal Graphs

Abstract

Let G be a graph with order n and adjacency matrix ${\textbf{A}}(G)$. The adjacency polynomial of G is defined as $\phi (G;\lambda ) =\det (\lambda {\textbf{I}}-{\textbf{A}}(G))=\sum _{i=0}^n\mathbf {a_{\textit{i}}}(G)\lambda ^{n-i}$. Here, ${\textbf{a}}_i(G)$ is referred as the i-th adjacency coefficient of G. We use ${\mathfrak {G}}_{n,m}$ to denote the set of all connected graphs having n vertices and m edges. A graph $G\in {\mathfrak {G}}_{n,m}$ is said to be 4-Sachs optimal in ${\mathfrak {G}}_{n,m}$ if

$$\begin{aligned} {\textbf{a}}_4(G)=\min \{{\textbf{a}}_4(H)|H\in {\mathfrak {G}}_{n,m}\}. \end{aligned}$$

The minimum 4-Sachs number in ${\mathfrak {G}}_{n,m}$ is denoted by $\overline{{\textbf{a}}_4}({\mathfrak {G}}_{n,m})$, which is the value of $\min \{{\textbf{a}}_4(H)|H\in {\mathfrak {G}}_{n,m}\}$. In this paper, we study the relationship between the value of ${\textbf{a}}_4(G)$ and the structural properties of G. Specially, we provide a structural characterization of 4-Sachs optimal graphs by showing that each 4-Sachs optimal graph in ${\mathfrak {G}}_{n,m}$ contains a connected difference graph as its spanning subgraph. Additionally, for $n\ge 4$ and $n-1\le m\le 2n-4$, we determine all 4-Sachs optimal graphs, along with their corresponding minimum 4-Sachs number $\overline{{\textbf{a}}_4}({\mathfrak {G}}_{n,m})$.