单模双环图
Unimodular Bicyclic Graphs
AI总结：
本文针对单模图的研究，在已完全刻画单模单环图的基础上，给出了单模双环图的完整刻画，并确定了双环图邻接矩阵行列式的所有可能取值。
AI中文摘要：
设G为具有邻接矩阵A(G)的简单无向图，若图G满足det A(G)∈{-1,1}，则称G为单模图。顶点数为m、边数为m+k-1的连通图称为k环图，特别地，双环图是顶点数为m、边数为m+1的图。单模单环图已被完全刻画，本文研究双环图对应的问题，给出单模双环图的完整刻画，并确定双环图G的det A(G)的所有可能取值。本研究的动机源于单模图在图逆理论中的核心作用，及其与特征值互反性和图的其他谱性质的关联。
英文摘要：
Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. A graph $G$ is said to be \emph{unimodular} if $\det A(G)\in\{-1,1\}$. A connected graph with $m$ vertices and $m+k-1$ edges is called \emph{$k$-cyclic}; in particular, a bicyclic graph has $m$ vertices and $m+1$ edges. Unimodular unicyclic graphs have been completely characterized. In this paper, we investigate the corresponding problem for bicyclic graphs. We provide a complete characterization of unimodular bicyclic graphs and determine all possible values of $\det A(G)$ for a bicyclic graph $G$. Our study is motivated by the central role of unimodular graphs in the theory of graph inverses and their connections with eigenvalue reciprocity and other spectral properties of graphs.
