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.
Let $G$ be a connected graph, and let $\lambda_1(G)>\lambda_2(G)$ denote its two largest adjacency eigenvalues. The spectral gap of $G$ is defined as the difference $\lambda_1(G) - \lambda_2(G)$. For integers $r\geq 2$ and $s\geq 0$, the double kite $DK(r,s)$ is formed by taking two vertex-disjoint copies of the comple...
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
A graph $G$ is det-extremal if $|\operatorname{det} A|=\operatorname{per} A$ for its adjacency matrix $A$. Det-extremal cubic bipartite graphs arise in the study of P\'olya's permanent problem, and McCuaig characterized the $3$-connected ones as vertex-sums of copies of the Heawood graph. The total domatic number of a...
A full-homomorphism from a graph $G$ to a graph $H$ is a function on vertex sets that preserves adjacency and non-adjacency of vertices. A graph $G$ is called a minimal $H$-obstruction if it has no full-homomorphism to $H$ but every proper vertex induced subgraph of $G$ does. Such graphs can have at most $|V(H)|+1$ ver...
Z. Rahimi, M. H. Shirdareh-Haghighi, Asma Namazi Department of Mathematics et al.· 0 citations
For a graph $G$, let avm($G$) denote the average size of its maximal matchings. Engbers and Erey initiated the extremal study of this parameter and asked for extensions from trees and unicyclic graphs to $k$-cyclic graphs. In this paper, we determine the maximum value of avm($G$) over all connected bicyclic graphs with...
Kainan Zhang· 0 citations
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