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arXiv2026

Unimodular Bicyclic Graphs

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.

math.CO↗

arXiv2026

On the singularity and the inverse of 3-colored digraphs

This article considers the class of connected 3-colored digraphs. Let $G$ be a 3-colored digraph and $A(G)$ be its adjacency matrix. $G$ is said to be non-singular (resp. singular) if $A(G)$ is a non-singular (resp. singular) matrix. A connected digraph is k-cyclic if it has $n$ vertices and $n+k-1$ edges. The main objective of this article is to provide a characterization of non-singular 3-colored unicyclic and bicyclic digraphs. If $A(G)$ is non-singular and $A(G)^{-1}$ has a $zero$ diagonal, then $A(G)^{-1}$ can be realized as the adjacency matrix of a digraph with complex weights. Therefore, we also identify all 3-colored bicyclic digraphs such that the diagonal of $A(G)^{-1}$ is zero. Furthermore, we study the invertibility of these digraphs and identify all those bicyclic 3-colored digraphs whose inverse is also a 3-colored digraph. We conduct the same study for the class of unicyclic 3-colored digraphs.

math.CO↗