On Bipartite Graphs Having Minimum Fourth Adjacency Coefficient

Abstract

Let G be a simple graph with order n and adjacency matrix ${\mathbf {A}}(G)$. The characteristic polynomial of G is defined by $\phi (G; \lambda )=\det (\lambda I-{\mathbf {A}}(G))=\sum _{i=0}^n{\mathbf {a}}_i(G)\lambda ^{n-i}$, where ${\mathbf {a}}_i(G)$ is called the i-th adjacency coefficient of G. Denote by ${\mathfrak {B}}_{n,m}$ the collection of all connected bipartite graphs having n vertices and m edges. A bipartite graph G is referred as 4-Sachs optimal if

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

For any given integer pair (n, m), in this paper we investigate the 4-Sachs optimal bipartite graphs. Firstly, we show that each 4-Sachs optimal bipartite graph is a difference graph. Then we deduce some structural properties on 4-Sachs optimal bipartite graphs. Especially, we determine the unique 4-Sachs optimal bipartite (n, m)-graphs for $n\ge 5$ and $n-1\le m\le 2(n-2)$. Finally, we provide a method to construct a class of cospectral difference graphs, which disprove a conjecture posed by Andelić et al. (J Czech Math 70:1125–1138, 2020).