S. Pirzada, T. A. Naikoo, F. A. Dar
The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.
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A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of integers is the signed degree set of some connected signed bipartite graph. \medskip \medskip
The net-regular strongly regular signed graphs with degree 5
该研究针对度为5的连通净正则强正则符号图开展分类工作，明确了净度为3的有5个、净度为1的有2个，完善了相关符号图的分类结果。
For positive integers $n,r,k$ with $n\ge r$ and $k\ge2$, a set $\{(x_1,y_1),(x_2,y_2),\dots,(x_r,y_r)\}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,\dots,x_r$ are distinct elements of $[n]$ and $y_1\dots,y_r\in[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $t\ge2$, extending a Hilton-Miln…
Degree similar and net-degree similar signed graphs
该研究将度相似概念扩展至符号图，证明符号树由度相似类唯一确定但不由净度相似类确定，构造了（净）度相似符号图，解决了Godsil和Sun关于单圈图问题的净度版本。
Orthogonal signed graphs of degree 5
本文完成了最大度为5的正交符号图的分类，并构造了6正则和8正则正交符号图的新的无限族，推进了该领域的研究。
The net-regular strongly regular signed graphs with degree 6
该研究针对度为6的净正则强正则符号图展开，确定了所有连通的6-正则净正则强正则符号图，并明确了不同净度对应的图的数量。
Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.
New constructions of signed difference sets
该研究针对带符号差集的构造问题，利用部分差集、乘积方法与分圆类推导得到新的存在性结果，为其在通信与编码理论中的应用提供理论支撑。
A signed graph is a graph with a function that assigns a label of positive or negative to each edge. The sign of a circle is the product of the signs of its edges; a graph is balanced if all of its circles are positive. A set of edges whose negation yields a balanced graph is a negation set. Results: tests to determine whether a negation set is minimal, minimum, or the unique minimum; any two disjoint negation sets must be bipartite; two classes of graphs are shown to have bipartite negation sets (in general, existence is an unsolved problem); I give an algorithm which finds a maximum family o…
A $(v,k,λ)$ difference set in a group $G$ of order $v$ is a subset $\{d_1, d_2, \ldots,d_k\}$ of $G$ such that $D=\sum d_i$ in the group ring $\mathbb{Z}[G]$ satisfies $$D D^{-1} = n + λG,$$ where $n=k-λ$. If $D=\sum s_i d_i$, where the $s_i \in \{ \pm 1\}$, satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all $s_i=1$) and circulant weighing matrices ($G$ cyclic and $λ=0$). We will show that there are other cases of interest, and give some results on their existence.
A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by the number of negative edges incident to $v$. Finally, we say $G$ realizes the set $D$ if: $$ D = \{sdeg(v) \text{ : } v\in V(G) \}. $$ The topic of signed degree sets and signed degree sequences has been studied from many directions. In this paper, we study properties needed for signed trees to have a given signed degree set. We start by proving that $D$ is …
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…
A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$ satisfies $\f(v)\not= \sg(vw)\f(w)$, where $\sg(vw)$ is the sign of the edge $vw$. For an integer $h\geq 0$, let $\Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\}$ and $\Ga_{2h+1}=\Ga_{2h} \cup \{0\}$. Following \cite{MaRS2015}, the signed chromatic number $\scn(G)$ of $G$ is the least integer $k$ such that $G$ admits a vertex coloring $\f$ with ${\rm im}(\f)\subseteq \Ga_k$…
Nonexistence results and constructions for signed difference sets
该研究针对有限阿贝尔群中的符号差集，建立了新的非存在性准则与存在性构造，排除了大量开放案例并解决了部分开放问题。
We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks'-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the under…
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.
Chordal signed graphs and signed bigraphs
本文研究了弦符号图和弦符号双图的特征化性质，揭示了其与严格弦有向图的关系，并给出了禁止子图的特征化方法。
Descent sets of cyclic permutations in types B and D
本研究针对B型带号对称群构造了对应Elizalde双射的函数Φ，证明其在D型循环带号置换及补集上的限制为双射，且可退化为原双射，还得到了循环带号置换的下降与标志主指标统计量的渐近正态性。
Signed Total Roman Domination and Domatic Numbers: Degree Three and Complete Multipartite Graphs
研究三次图的符号全罗马支配数与其开放包装数、2-元全支配数和符号全支配数的关系，得到紧界和NP完全性结果，并确定完全多部图的精确值。
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 $mad(G)$ …
Let $k, r, n \geq 1$ be integers, and let $§_{n, k, r}$ be the family of $r$-signed $k$-sets on $[n] = \{1, \dots, n\}$ given by $$ \mathcal{S}_{n, k, r} = \Big\{\{(x_1, a_1), \dots, (x_k, a_k)\}: \{x_1, \dots, x_k\} \in \binom{[n]}{k}, a_1, \dots, a_k \in [r] \Big\}. $$ A family $\mathcal{A} \subseteq §_{n, k, r}$ is \emph{intersecting} if $A, B \in \mathcal{A}$ implies $A \cap B \not= \emptyset$. A well-known result (first stated by Meyer and proved using different methods by Deza and Frankl, and Bollobás and Leader) states that if $\mathcal{A} \subseteq \mathcal{S}_{n, k, r}$ is intersectin…
Fractional balanced chromatic number and arboricity of planar (signed) graphs
本文构造平面符号图，证明其分数平衡色数可大于2，反驳了平面图分数树状度上界为2的猜想，并给出色数上确界至少为83/41、树状度极限为2+2/25的序列。
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 signed graph $Σ$ is a pair $(G,σ)$, where $G=(V,E)$ is the underlying graph in which each edge is assigned $+1$ or $-1$ by the signature function $σ:E\rightarrow\{-1,+1\}$. In this paper, we extend the extensively applied concepts of metric dimension and resolving sets for unsigned graphs to signed graphs. We analyze the metric dimension of some well known classes of signed graphs including a special case of signed trees. Among other things, we establish that the metric dimension of a signed graph is invariant under negation.
The Erdős--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.
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
本文研究符号图的列表边着色，将经典列表边着色与符号边着色推广，证明树宽为3的符号图及树宽4且Δ≥10的符号图满足 Vizing 猜想的符号类似。
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.
We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm λ$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm λ$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any d-dimensional simplex in general position into d! signed sets, each of which corresponds to a permutation in the symmetric group, and reduce the problem of counting lattice points in a polytope in general position to that of counting lattice points in these special signed sets. Applying this decomposition to a lattice-face simplex, we obtain signed sets with …
Signed permutations and degree-one dot action representations for types B and C
本文研究B/C型符号置换Weyl群的样条环$\mathcal{M}_{H}$，通过其分次结构与点作用表示，给出一次部分的显式生成元并利用$H$的组合数据计算两种点作用表示的一次部分，实现A型相关结构的B/C型推广。
Vector Valued Switching in Signed Graphs
本文提出带符号图的向量值切换函数概念，定义平衡维度与强平衡维度以分类不平衡程度，并给出其界的估计与部分图类的计算。
Let n>0 be an integer and let B_{n} denote the hyperoctahedral group of rank n. The group B_{n} acts on the polynomial ring Q[x_{1},...,x_{n},y_{1},...,y_{n}] by signed permutations simultaneously on both of the sets of variables x_{1},...,x_{n} and y_{1},...,y_{n}. The invariant ring M^{B_{n}}:=Q[x_{1},...,x_{n},y_{1},...,y_{n}]^{B_{n}} is the ring of diagonally signed-symmetric polynomials. In this article we provide an explicit free basis of M^{B_{n}} as a module over the ring of symmetric polynomials on both of the sets of variables x_{1}^{2},..., x^{2}_{n} and y_{1}^{2},..., y^{2}_{n} usi…
Vizing's theorem for signed multigraphs
证明带符号多重图的色指数不超过最大度与最大重数之和，推广了Vizing定理和Behr定理。
Weakly Negative Circles Versus Best Clustering in Signed Graphs
本文研究符号图聚类中违反规则的最小边数 Q 与边不相交弱负圈最大数 w 的关系，并刻画了满足 Q=w 的符号图结构。
Gremban Expansion for Signed Networks: Algebraic and Combinatorial Foundations for Community-Faction Detection
本文通过Gremban扩张将符号图提升为无符号图，建立了尊重对称的割集与符号图正则割集和挫败集的双射对应，提出基于谱聚类的社区-派系检测新方法。
Recently Naserasr, Sopena, and Zaslavsky [R. Naserasr, É. Sopena, T. Zaslavsky,Homomorphisms of signed graphs: An update, arXiv: 1909.05982v1 [math.CO] 12 Sep 2019.] published a report on closed walks in signed graphs. They gave a characterization of the sets of closed walks in a graph $G$ which corespond to the set of negative walks in some signed graph on $G$. In this note we show that their characterization is not valid and give a new characterization.
In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound.
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph $(G,σ)$ to a signed graph $(H, π)$ to be a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ which preserves incidence, adjacency and the signs of closed walks. In this work we first give a characterization of the s…
Let $1 \leq k \leq n$ be a positive integer. A {\em nonnegative signed $k$-subdominating function} is a function $f:V(G) \rightarrow \{-1,1\}$ satisfying $\sum_{u\in N_G[v]}f(u) \geq 0$ for at least $k$ vertices $v$ of $G$. The value $\min\sum_{v\in V(G)} f(v)$, taking over all nonnegative signed $k$-subdominating functions $f$ of $G$, is called the {\em nonnegative signed $k$-subdomination number} of $G$ and denoted by $γ^{NN}_{ks}(G)$. When $k=|V(G)|$, $γ^{NN}_{ks}(G)=γ^{NN}_s(G)$ is the {\em nonnegative signed domination number}, introduced in \cite{HLFZ}. In this paper, we investigate seve…
A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is called sign-unbalanced. A nut graph has a one dimensional kernel with a corresponding eigenvector that is full. In this paper we generalise the notion of nut graphs to signed graphs. Orders for which unsigned regular nut graphs exist were determined recently for the degrees up to $11$. By extending the definition to signed nut graphs, we find all pairs $(ρ, n)$ for which a $ρ$-regular nut graph (sign-balanced…
Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The \emph{signed $k$-domination number} of $G$ is the minimum value of $\sum_{v\in V}f(v)$ taken over all signed $k$-dominating functions of $G$. The \emph{signed total $k$-dominating function} and \emph{signed total $k$-domination number} of $G$ can be similarly defined by changing the closed neighborhood $N_G[v]$ to the open neighborhood $N_G(v)$ in the definit…
For a simple signed graph $G$ with the adjacency matrix $A$ and net degree matrix $D^{\pm}$, the net Laplacian matrix is $L^{\pm}=D^{\pm}-A$. We introduce a new oriented incidence matrix $N^{\pm}$ which can keep track of the sign as well as the orientation of each edge of $G$. Also $L^{\pm}=N^{\pm}(N^{\pm})^T$. Using this decomposition, we find the numbers of positive and negative spanning trees of $G$ in terms of the principal minors of $L^{\pm}$ generalizing Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix $Q^{\pm}=D^{\pm}+A$ along w…
Index perturbation of signed graphs
研究符号图删除顶点后最大特征值的变化，给出上界并刻画极图。
Spherical two-distance sets and eigenvalues of signed graphs
本研究聚焦高维固定双角度球面双距离集最大规模问题，提出极限值与符号图特征值重数相关的猜想，确定了两类参数下的极限值，是等角线外首个非平凡参数下的极限确定成果。
We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source…
A finite dynamical system with $n$ components is a function $f:X\to X$ where $X=X_1\times\dots\times X_n$ is a product of $n$ finite intervals of integers. The structure of such a system $f$ is represented by a signed digraph $G$, called interaction graph: there are $n$ vertices, one per component, and the signed arcs describe the positive and negative influences between them. Finite dynamical systems are usual models for gene networks. In this context, it is often assumed that $f$ is {\em degree-bounded}, that is, the size of each $X_i$ is at most the out-degree of $i$ in $G$ plus one. Assumi…
We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility $τ$ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Fina…
An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs.
We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph…
A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention …
Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $π$ in $S_n$ such that for any $1\leq j \leq n$, $\{π(1),π(2),\dots,π(j)\}$ is an interval in $\mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and gener…
A circular $r$-coloring of a signed graph $(G, σ)$ is an assignment $ϕ$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G, σ)$ such that for each positive edge $uv$ of $(G, σ)$ the distance of $ϕ(v)$ and $ϕ(v)$ is at least 1 and for each negative edge $uv$ the distance of $ϕ(u)$ from the antipodal of $ϕ(v)$ is at least 1. The circular chromatic number of $(G, σ)$, denoted $χ_c(G, σ)$, is the infimum of $r$ such that $(G, σ)$ admits a circular $r$-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed $d$-d…
A sign is introduced in the usual Laplacian on graphs and the corresponding analogue of the isoperimetric constant for this Laplacian is presented, i.e. a geometric quantity which enables to bound from above and below the first eigenvalue. The introduction of the sign in the Laplacian is motivated by the study of $2$-lifts of graphs and of the combinatorial Laplacian in higher degree.
A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd edges and the other edges of $E$ even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must be an odd edge and an even edge connecting $i$ and $j$, or there are no edges connect…
In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $Γ$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $Γ$ of order $k$ with $ε(Γ)=d$, where $ε(Γ)$ is the largest integer $d$ for which $Γ$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We focus on the combinatorial structure of $Γ$-flows in a signed gra…
Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs
本文研究有向图的最大特征值和拉普拉斯特征值之和的界，证明了非二部图包含平衡三角形的条件，并验证了一个关于拉普拉斯特征值与顶点度数关系的猜想。
The nullity of the net Laplacian matrix of a signed graph
本文研究连通符号图的净拉普拉斯矩阵零度，证明了其上下界，并完全确定了零度达到边界时的连通符号图与符号仙人掌图的结构特征。
The notion of corona of two graphs was introduced by Frucht and Harary in 1970. In this paper we generalize their definition of corona product of two graphs and introduce corona product of two signed graphs by utilizing the framework of marked graphs, which was introduced by Beineke and Harary in 1978. We study the structural and spectral properties of corona product of signed graphs. Further we define signed corona graphs by considering corona product of a fixed small signed graph with itself iteratively, and we call the small graph as the seed graph for the corresponding corona graphs. Signe…
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