Signed graphs with exactly two main eigenvalues: The unicyclic case
Shanxi University（山西大学）；Changan University（长安大学）；Hubei Normal University（湖北师范大学）；South China Normal University（华南师范大学）；Shennong Technology Group Co., Ltd.（神农科技集团有限公司）
Zenan Du, Fenjin Liu, Hechao Liu, Jifu Lin, Wenxu Yang
An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.
Let $x_1, x_2, \dots, x_n$ be the eigenvalues of a signed graph $Γ$ of order $n$. The energy of $Γ$ is defined as $E(Γ)=\sum^{n}_{j=1}|x_j|.$ Let $\mathcal{P}_n^4$ be obtained by connecting a vertex of the negative circle $(C_4,{\overlineσ})$ with a terminal vertex of the path $P_{n-4}$. In this paper, we show that for $n=4,6$ and $n \geq 8,$ $\mathcal{P}_n^4$ has the maximal energy among all connected unicyclic $n$-vertex signed graphs, except the cycles $C_5^+, C_7^+.$
Let $Γ=(K_{n},H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. The index of $Γ$ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of $Γ$ when $H$ is a unicyclic graph. We show that among all signed complete graphs of order $n>5$ whose negative edges induce a unicyclic graph of order $k$ and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle.
Given a graph $G$, the burning number of $G$ is the smallest integer $k$ for which there are vertices $x_1, x_2,\ldots,x_k$ such that $(x_1,x_2,\ldots,x_k)$ is a burning sequence of $G$. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A $t$-unicyclic graph is a unicycle graph with exactly one vertex of degree greater than $2$. In this paper, we first present the bounds for the burning number of $t$-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burn…
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,σ)$ is a weighted graph with a special weight function $σ: E(G)\to \{-1,1\}$. A graph is sign-invertible (or sign-invertible) if its inverse is a signed graph. A sign-invertible graph is always unimodular. The inverses of graphs have interesting combinatorial interests. In this paper, we study inverses of graphs and provide a combinatorial description for sign-invertible graphs, whic…
The total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar definitions of the line signed graph, we define the corresponding total signed graph and we show that it is stable under switching. We consider balance, the frustration index and frustration number, and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency matrix of the total graph and the spectra of certain compositions, and we determine some with exactly two main eigenvalues.
We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz's characterization, the van Rooij and Wilf's characterization and the Beineke's characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than $-2$, or les…
A signed magic rectangle $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. In this paper we prove that a signed magic rectangle $SMR(m,n;r, 2)$ exists if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.
Colouring negative exact-distance graphs of signed graphs
Zhejiang Normal University（浙江师范大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息学研究所）；Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)（分析与数学社会中心）；Instituto de Ingeniería Matemática-CIMFAV（数学工程研究所-CIMFAV）；Universidad de Valparaíso（瓦尔帕莱索大学）；Computer Science Institute of Charles University（查理大学计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers critical groups over polynomial rings.
Lovász has completely characterised the structure of graphs with no two vertex-disjoint cycles, while Slilaty has given a structural characterisation of graphs with no two vertex-disjoint odd cycles; his result is in fact more general, describing signed graphs with no two vertex-disjoint negative cycles. A biased graph is a graph with a distinguished set of cycles (called balanced) with the property that any theta subgraph does not contain exactly two balanced cycles. In this paper we characterise the structure of biased graphs with no two vertex-disjoint unbalanced cycles, answering a questio…
A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the sensitivity conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this connection is a natural first example of the relationship between group labeled adjacency matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define a two-eigenvalue cover to be a covering graph whose adjacency spectra differs (as a multiset) from that of the graph it covers by exactly two eigenvalues. We show that…
