A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges of $E$ even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are adjacent and connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are adjacent and connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The parameters $M(G,Σ)$ and $ξ(G,Σ)$ of a signed graph $(G,Σ)$ are the largest nullity of any matrix $A\in S(G,Σ)$ and the largest nullity of any matrix $A\in S(G,Σ)$ that has the Strong Arnold Hypothesis, respectively. In a previous paper, we gave a characterization of signed graphs $(G,Σ)$ with $M(G,Σ)\leq 1$ and of signed graphs with $ξ(G,Σ)\leq 1$. In this paper, we characterize the $2$-connected signed graphs $(G,Σ)$ with $M(G,Σ)\leq 2$ and the $2$-connected signed graphs $(G,Σ)$ with $ξ(G,Σ)\leq 2$.
In this paper, an upper bound on the nullity of signed graphs in terms of the cyclomatic number and the number of pendant vertices is proved, and the corresponding extremal signed graphs are completely characterized.
Let $G^σ=(G,σ)$ be a signed graph and $A(G,σ)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $η(G,σ)$ be the nullity of $(G,σ)$. He et al. [Bounds for the matching number and cyclomatic number of a signed graph in terms of rank, Linear Algebra Appl. 572 (2019), 273--291] proved that $$|V(G)|-2m(G)-c(G)\leqη(G,σ)\leq |V(G)|-2m(G)+2c(G),$$ where $c(G)$ is the dimension of cycle space of $G$. Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that no signed graphs with nullity $|V(G)|…
The rank $r(G)$ of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the nullity $η(G)$ of $G$ is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this paper, we prove that if $G$ is a connected graph of order $n$ with rank $r$, then $G$ contains a nonsingular connected induced subgraph of order $r$. As an application of the result, we completely solve the following problem posed by Zhou, Wong and Sun in [Linear Algebra and its Applications, 555 (2018) 314-320]: Let $G$ be a connected graph of order $n$ with nullity $η(G)$ and the maximum degree $Δ$. Then $$η(G)\le\frac{(Δ-2)n+2…
For a simple graph $G=(V,E),$ let $\mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij}\neq 0$ if $\{i,j\}\in E$ and $a_{ij}=0$ if $\{i,j\}\notin E$. The maximum positive semidefinite nullity of $G$, denoted $\operatorname{M}_+(G),$ is $\max\{\operatorname{null}(A)|A\in \mathcal{S}_+(G)\}.$ A tree cover of $G$ is a collection of vertex-disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. The tree cover number of $G$, denoted $T(G)$, is the cardinality of a minimum tree cover. It is known that the tree …
A signed graph is a graph with a positive or negative sign on each edge. Regarding each edge as two half edges, an orientation of a signed graph is an assignment of a direction to each of its half edges such that the two half edges of a positive edge receive the same direction and that of a negative edge receive opposite directions. A signed graph with such an orientation is called a bidirected graph. A nowhere-zero $k$-flow of a bidirected graph is an assignment of an integer from $\{-(k-1), \ldots, -1, 1, \ldots, (k-1)\}$ to each of its half edges such that Kirchhoff's law is respected, that…
A signed graph is a graph $G$ associated with a mapping $σ: E(G)\to \{-1,+1\}$, denoted by $(G,σ)$. A $cycle$ of $(G,σ)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges, and negative otherwise. A $circuit$ of of a signed graph $(G,σ)$ is a positive cycle or a barbell consisting of two edge-disjoint negative cycles joined by a path. The definition of a circuit of signed graph comes from the signed-graphic matroid. A circuit cover of $(G,σ)$ is a family of circuits covering all edges of $(G,σ)$. A circuit cover with the smallest total lengt…
A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $ϕ$ is an $A$-flow in $G$, then let $N(ϕ)$ be the nullity of $ϕ$, that is, the set of edges $e$ of $G$ with $ϕ(e)=0$. In this paper we show that (a) Tutte's 5-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $3$-edge-connected cubic graphs admit a $\mathbb{Z}_5$-flow $ϕ$ (not necessarily no-where zero), such that $|N(ϕ)|\leq f(|E(G)|)$; (b) Tutte's 4-flow conjecture is equivalent to the statement that …
A signed graph is a graph in which each edge has a positive or negative sign. In this article, first we characterize the distance compatibility in the case of a connected signed graph and discussed the distance compatibility criterion for the cartesian product, lexicographic product and tensor product of signed graphs. We also deal with the distance matrix of the cartesian product, lexicographic product and tensor product of signed graphs in terms of the distance matrix of the factor graphs.
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.
A signed circuit is a minimal signed graph (with respect to inclusion) that admits a nowhere-zero flow. We show that each flow-admissible signed graph on $m$ edges can be covered by signed circuits of total length at most $(3+2/3)\cdot m$, improving a recent result of Cheng et al. [manuscript, 2015]. To obtain this improvement we prove several results on signed circuit covers of trees of Eulerian graphs, which are connected signed graphs such that removing all bridges results in a collection of Eulerian graphs.
In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The proof is based on what we call Hoffman's limit theorem which we formulate for Hermitian matrices, and also the extension of the concept of Hoffman graph and line graph for the setting of signed graphs.
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.
Spanning trees and even integer eigenvalues of graphs
对于图$G$，设$L(G)$和$Q(G)$分别是$G$的拉普拉斯矩阵和无符号拉普拉斯矩阵，$τ(G)$是$G$的生成树个数.我们证明了，如果$G$有一个奇数的顶点和$τ(G)$是不可分的$4$，那么（i）$L(G)$没有偶数本征值，（ii）$Q(G)$没有整数本征值$λ\equiv2\pmod4$，和（iii）$Q(G)$最多有一个本征值$λ\equiv0\pmod4$，这样的本征值是简单的。作为一个结果，我们扩展了以前的结果由古特曼和Sciriha和Bapat的线性图的邻接矩阵的零。我们还证明了如果$τ(G)=2^ts$的$s$是奇数，则$Q(G)$的任何偶数特征值的重数至多是$t+1$ .除其他事项外，我们证明，如果$L(G)$或$Q(G)$有一个偶数特征值的重至少$2$，那么$τ(G)$是可分的$4$。作为这一结果的一个非常特殊的情况，Zhou等[On the nullity of connected graphs with least energy at least $-2$，Appl. Anal. 7（2013），250- 261]关于单圈图的线图的邻接矩阵的零性的讨论如下。
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any $k\geq 5$ there are infinitely many connected signed $k$-regular graphs having maximum eigenvalue $\sqrt{k}$. Moreover for each $m\ge…
In this paper, we consider the Randić energy $RE$ of simple connected graphs. We provide upper bounds for $RE$ in terms of the number of vertices and the nullity of the graph. We present families of graphs that satisfy the Conjecture proposed by Gutman, Furtula and Bozkurt \cite{Gutman1} about the maximal RE. For example, we show that starlikes of odd order satisfy the conjecture.
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 $C_5^+, C_7^+.$
The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, σ)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, σ)$ admits a circular $r$-coloring is said to be the circula…
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…
A signed graph $ (G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, Σ')$ by re-signing some vertices of $ (G, Σ)$, then $ (G, Σ) \equiv (G, Σ')$. A signed graphs $ (G, Σ)$ admits an homomorphism to $ (H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $ (G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signe…
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 structure theory for signed graphs with fixed smallest eigenvalue
University of Science and Technology of China（中国科学技术大学）；CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China（中国科学院吴文俊数学重点实验室，中国科学技术大学）；Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学新拓数学中心）
Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bidegeless subgraph of $G$. Further, we are able to use this result to show that every flow-admissible signed graph $(G,σ)$…
In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, σ)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, σ)$ with no positive bridge, denoted $Φ_c(G,σ)$, is the minimum $r$ such that $(G, σ)$ admits a circular $r$-flow. This is the dual notion of circular c…
Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as to model the spread of diseases and information in social networks. In this paper, we introduce and study the connected forcing process -- a restriction of zero forcing in which the initially colored set of vertices induces a connected subgraph. We show that the connected forcing number -- the cardinality of the smallest initially colored vertex set which f…
Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices
University of Science and Technology of China（中国科学技术大学）；CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China（中国科学院吴文俊数学重点实验室，中国科学技术大学）；Shanghai University（上海大学）
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…
A signed graph $Σ=(G,σ)$ is said to be parity signed if there exists a bijection $f : V(G) \rightarrow \{1,2,...,|V(G)|\}$ such that $σ(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity, where $uv$ is an edge of $G$. The rna number of a graph $G$, denoted $σ^{-}(G)$, is the minimum number of negative edges among all possible parity signed graphs over $G$. The rna number is also equal to the minimum cut size that has nearly equal sides. In this paper, for generalized Petersen graph $P(n,k)$, we prove that $3 \leq σ^{-}(P(n,k)) \leq n$ and these bounds are sharp. The exact value of $σ^{…
