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.
In this paper, we focus on the index ( largest eigenvalue) of the adjacency matrix of connected signed graphs. We give some general results on the index when the corresponding signed graph is perturbed. As applications, we determine the first five largest index among all unbalanced bicyclic graphs on n >= 36 vertices together with the corresponding extremal signed graphs whose index attain these values.
The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the corrected version of) a conjecture by Koledin and Stanić (2017).
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.
The Wiener index of a graph $W(G)$ is a well studied topological index for graphs. An outstanding problem of Šolt{é}s is to find graphs $G$ such that $W(G)=W(G-v)$ for all vertices $v\in V(G)$, with the only known example being $G=C_{11}$. We relax this problem by defining a notion of Wiener indices for signed graphs, which we denote by $W_σ(G)$, and under this relaxation we construct many signed graphs such that $W_σ(G)=W_σ(G-v)$ for all $v\in V(G)$. This ends up being related to a problem of independent interest, which asks when it is possible to $2$-color the edges of a graph $G$ such that …
We introduce the concept of deficiency in signed graphs. The deficiency of a coloration is the number of unused colors. We classify the deficiency of 2-chromatic graphs. There are four decision problems about the minimum and maximum deficiency of a 3-chromatic signed graph. We answer two of them with a polynomial-time algorithm for deciding the maximum deficiency of a 3-chromatic signed graph.
School of Mathematics and Statistics,Northwestern Polytechnical University（西北工业大学数学与统计学院）；Xi’an-Budapest Joint Research Center for Combinatorics,Northwestern Polytechnical University（西北工业大学西安-布达佩斯组合学联合研究中心）
A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ such that, after switching some of the vertices of $G$, $φ$ maps every edge of $G$ to an edge of $H$ of the same type. The chromatic number $χ_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. The maximum average degree $mad(G)$ …
A homomorphism of a signed graph $(G, σ)$ to $(H, π)$ is a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ such that adjacencies, incidences and the product of signs of closed walks are preserved. Motivated by reformulations of the $k$-coloring problem in this language, and specially in connection with results on $3$-coloring of planar graphs, such as Grötzsch's theorem, in this work we consider bounds on maximum average degree which are sufficient for mapping to the signed graph $(K_{2k}, σ_m)$ ($k\geq 3$) where $σ_m$ assigns to edges of a perfect matching the…
A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph $G$ to a 2-edge-colored graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ that maps every edge in $G$ to an edge of the same type in $H$. Switching a vertex $v$ of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to $v$. There is a homomorphism from the signed graph $G$ to the signed graph $H$ if after switching some subset of the vertices of $G$ there is a 2-edge-colored homomorphism from $G$ to $H$. The chromatic number of a 2-e…
Signed list edge coloring in graphs of bounded treewidth
Yan’an University（延安大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用重点实验室（教育部））；School of Marine Science and Technology, Northwestern Polytechnical University（西北工业大学海洋科学与技术学院）
Turán-type extremal problems for unbalanced signed graphs
Northwestern Polytechnical University（西北工业大学）；Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University（西安-布达佩斯组合学联合研究中心）；School of Mathematical Sciences & Academy for Multidisciplinary Studies, Captial Normal University（首都师范大学数学科学学院与跨学科研究院）
Signed graphs have their edges labeled either as positive or negative. $ρ(M)$ denote the $M$-spectral radius of $Σ$, where $M=M(Σ)$ is a real symmetric graph matrix of $Σ$. Obviously, $ρ(M)=\mbox{max}\{λ_1(M),-λ_n(M)\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $(K_n,H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum $ρ(A(Σ))$ among $(K_n,T^-)$ where $T$ is a spanning tree? To answer the problem, we characterize the extremal signed grap…
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.
Spectral Turán problem for $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs
School of Mathematics and Statistics,Northwestern Polytechnical University（西北工业大学数学与统计学院）；Xi’an-Budapest Joint Research Center for Combinatorics,Northwestern Polytechnical University（西北工业大学西安-布达佩斯组合学联合研究中心）
A signed graph $(G,Σ)$ is a graph $G$ together with a set $Σ\subseteq E(G)$ of negative edges. A circuit is positive if the product of the signs of its edges is positive. A signed graph $(G,Σ)$ is balanced if all its circuits are positive. The frustration index $l(G,Σ)$ is the minimum cardinality of a set $E \subseteq E(G)$ such that $(G-E,Σ-E)$ is balanced, and $(G,Σ)$ is $k$-critical if $l(G,Σ) = k$ and $l(G-e, Σ- e)<k$, for every $e \in E(G)$. We study decomposition and subdivision of critical signed graphs and completely determine the set of $t$-critical signed graphs, for $t \leq 2$. Crit…
Kirchhoff-type Laws for signed graphs are characterized by generalizing transpedances through the incidence-oriented structure of bidirected graphs. The classical $2$-arborescence interpretation of Tutte is shown to be equivalent to single-element Boolean classes of reduced incidence-based cycle covers, called contributors. A generalized contributor-transpedance is introduced using entire Boolean classes that naturally cancel in a graph; classical conservation is proven to be property of the trivial Boolean classes. The contributor-transpedances on signed graphs are shown to produce non-conser…
