Amelia R. W. Mattern
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.
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We introduce joins of signed graphs and explore the chromatic number of the all-positive and all-negative joins. We prove an analogue to the theorem that the chromatic number of the join of two graphs equals the sum of their chromatic numbers. Given two signed graphs, the chromatic number of the all-positive and all-negative join is usually less than the sum of their chromatic numbers, by an amount that depends on the new concept of deficiency of a signed-graph coloration.
On the Cordial Deficiency of Complete Multipartite Graphs
计算了完全多部图的亲切边亏数，并给出了其亲切点亏数的一个上界。给出了两个亲切图的张量积亲切的条件。
Cyclic Deficiency of Graphs
如果图$G$的每个顶点$v$的与$v$关联的边被连续的颜色着色，且颜色$1$与颜色$t$连续，则称图$G$的正常边着色为循环区间$t$ -着色.在本文中，我们引入并研究了一个新的概念，一个图$G$的循环亏损，定义为最小数量的悬挂边的附件$G$产生一个图允许循环区间染色，这个数字可以被认为是一个衡量的接近$G$是循环区间可着色。我们确定或界定了几个图族的循环亏数。特别是，我们提出了有界最大程度的任意大的循环不足，和图的循环不足的顶点数的方法的图的例子。最后，我们猜想任何图的循环亏不超过顶点数，我们提出了支持这个猜想的几个结果。
Consistency in the Naturally Vertex-Signed Line Graph of a Signed Graph
一个带符号的图是一个边带符号的图。在顶点符号图中，顶点是符号的。如果每个圆中的符号的乘积都是正数，则称后者为相容的。有符号图的线图自然是顶点有符号的。基于Acharya，Acharya，和Sinha在2009年的一个刻画，我们给出了自然点符号线图一致的符号单图的构造.
A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs.
Signed degree sets in signed graphs
一个符号图G中顶点的不同符号度的集合D称为它的符号度集。本文证明了正（负）整数的每一个非空集都是某个连通符号图的符号度集，并确定了这类符号图的最小可能阶。我们还证明了每一个整数的非空集都是某个连通符号图的符号度集。
Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges a…
Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae for the distance spectrum of some unbalanced signed graphs. We also introduce the notion of distance-compatible signed graphs and partially characterize it.
In 2015, Matthias Beck and his team developed a computer program in SAGE which efficiently determines the number of signed proper $k$-colorings for a given signed graph. In this article, we determine the number of different signatures on Book graph up to switching isomorphisms. We also find a recursive formula of the signed chromatic polynomials of signed Book graphs.
A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.
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.
We construct infinitely many signed graphs having symmetric spectrum, by using the NEPS and rooted product of signed graphs. We also present a method for constructing large cospectral signed graphs. Although the obtained family contains only a minority of signed graphs, it strengthen the belief that the signed graphs with symmetric spectrum are deeper than bipartite graphs, i.e the unsigned graphs with symmetric spectrum.
A signed graph is an ordered pair $Σ=(G,σ),$ where $G=(V,E)$ is the underlying graph of $Σ$ with a signature function $σ:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an $n^{th}$ power of a signed graph to be unique. Also, we characterize balanced power signed graphs.
Cordial Deficiency
我们引入了两个新的图的非亲切性的措施。然后，我们计算这些措施的各种家庭的noncordial图的值。我们还确定了哪些莫比乌斯梯是亲切的。
Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.
Signed degree sets in signed bipartite graphs
一个带符号的二部图G（U，V）是一个二部图，其中每个边都被赋予一个正号或一个负号. G（U，V）中顶点x的符号度是与x关联的正边数减去与x关联的负边数。G（U，V）的顶点的不同符号度的集合S称为它的符号度集。本文证明了每一个整数集都是某个连通的符号二部图的符号度集。\medskip \medskip
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。
We describe parity labelings of signed graphs; equivalently, cuts of the underlying graph that have nearly equal sides. We characterize the balanced signed graphs which are parity signed graphs. We give structural characterizations of all parity signed stars, bistars, cycles, paths and complete bipartite graphs. The rna number of a graph is the smallest cut size that has nearly equal sides; we find it for a few classes of parity signed graphs.
We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations over distinguished order-ideals of the signed partition lattice. We also define the quotient of a signed graph by a signed permutation, and show that its signed graphic arrangement is closely related to an induced arrangement on a distinguished subspace. Lastly, a formula for the number of unlabeled acyclic orientations of a signed graph is pr…
Critically 3-frustrated signed graphs
本文研究临界挫败符号图，猜想不可分解且非细分得到的临界k-挫败符号图有限，并证明临界3情形下的两类有限分类结果及构造方法。
In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed…
Spectral integral variation of signed graphs
本文研究了带符号图在添加新边后带符号拉普拉斯矩阵的谱变差为整数的条件，并完全刻画了可递归添加新边并保持谱积分变差以生成给定带符号完全图的带符号图类。
A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioabă, Koolen, and Wang (2018).
Choosability in signed planar graphs
研究了符号可平面图的可选择性。证明了每个带符号的可平面图都是5-可选的，并且存在一个带符号的可平面图不是4-可选的，而无符号的图是4-可选的.对每个$k \in \{3,4,5,6\}$，每个不含长度为$k$的圈的带符号可平面图是4-可选的.此外，每个不含长为3和长为4的圈的带符号可平面图都是3-可选的。我们构造了一个围长为4的带符号可平面图，它不是3-可选图，而无符号图是3-可选图.
On token signed graphs
本文提出 k-token 符号图，证明切换同构保持性及平衡图 Laplacian 谱包含关系，并引入不平衡度参数，分析其与挫折指数的关系。
Local measures of interval edge-uncolorability
本文提出图的局部亏格和弱局部亏格，度量图偏离区间可染色的程度，证明其与亏格差异可任意大，并给出构造性结果与开放问题。
Shorter signed circuit covers of graphs
有符号回路是一个最小的有符号图（相对于包含），它允许一个无处零流。我们证明了$m$边上的每个流容许符号图可以被总长度至多为$(3+2/3)\cdot m$的符号回路覆盖，改进了Cheng等人的最近结果。[手稿，2015]。为了实现这一改进，我们证明了几个结果签署电路覆盖树的欧拉图，这是连接签署的图形，使删除所有的桥梁结果在一个集合的欧拉图。
In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero $6$-flow. Bouchet himself proved that such signed graphs admit nowhere-zero $216$-flows and Zyka further proved that such signed graphs admit nowhere-zero $30$-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow.
Further results on the nullity of signed graphs
图的零度是指图的谱中特征值为零的重数。一个带符号的图是一个图，它的每一条边都有一个符号。本文得到了符号图的特征多项式的系数定理，给出了两个关于有割点的符号图的零值的公式。作为上述结果的应用，我们研究了双圈符号图$Γ(\infty(p,q,l))$的零度，得到了不平衡双圈符号图的零度集，从而确定了双圈符号图的零度集.
Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based on the action of its automorphism group. As an application and computational results, we classify all the switching non-isomorphic signed graphs arising from the complete graph K5 and the generalized Petersen graph GP(7; 2). Moreover, some results on the automorphism groups of the target signed graphs are obtained.
Hajós-like theorem for signed graphs
本文设计了五种图运算，证明了每一个色数为$q$的带符号图都可以从全正完全图$(K_q,+)$中通过重复使用这些图运算得到。这个结果给出了Hajós定理的一个符号版本，强调了全正完全图在符号图类中的作用，就像在无符号图类中一样。
Mycielskian of Signed Graphs
本文定义符号图的Mycielskian，研究其平衡性、切换性质、色数关系及相关矩阵结构，并给出构造平衡Mycielskian的条件。
Characterization of Line-Consistent Signed Graphs
有符号边的图的线图带有顶点符号。一个点符号图是一致的，如果每个圈（圈，回路）都有正的点符号积。Acharya，Acharya和Sinha最近刻画了线一致符号图，即，边符号图，其线图与自然导出的顶点签名是一致的。他们的证明适用于Hoede对一致顶点符号图的相对困难的刻画。我们给出了一个简单的证明，不依赖于Hoede定理以及线一致的符号图的结构描述。
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arran…
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 $m\ge…
The nullity of unicyclic signed graphs
本文引入了符号图的零性概念，并给出了具有悬挂树的符号图的零性的一些结果。分别刻画了零度为n-2; n-3; n-4; n-5的n阶单圈符号图.
New concept of connection in signed graphs
在有符号图中，每条边都有一个符号，$+1$或$-1$。本文利用点间正负链的存在性，给出了符号图中连通度的一个新定义。我们证明了这个定义的一些结果和性质，如符号组件，符号关节顶点，和符号峡，我们比较相应的图和符号图形拟阵的性质。我们将我们的结果应用到没有正圈的符号图。对于每条边都是负的符号图，我们的性质变成了奇偶性。
Signed circuit $6$-covers of signed $K_4$-minor-free graphs
本文针对符号图的符号回路覆盖问题，证明了Fan猜想对无K₄子式的符号图成立，即每个可覆盖的无K₄子式符号图都具有符号回路6-覆盖，并说明6-覆盖是最优可能的。
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.
Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-zero $10$-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.
r-orientation of a signed graph and its application on coronae of signed graphs
本文引入反向定向（r-orientation）概念定义了符号图的冠图变体，利用符号coronal获得了正则符号图在r-定向下多种矩阵的特征多项式，并借此构造了无限多对共谱符号图族。
A signed graph product and its Application
该研究定义了一种新的带符号图乘积，推导了其及对应无符号图乘积的各类谱特性，确定了其为整谱图的条件，并构造出同谱与非同谱等能的带符号图序列。
The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary clique problem is in P. We also study the complexity of further related problems.
Nowhere-zero flows on signed supereulerian graphs
本文验证了Bouchet关于带符号图无处为零6-流的猜想在具有生成偶欧拉子图的图类上成立，并通过构造保号双射证明其紧性，且推广至阿贝尔凯莱图。
There are several ways to generalize graph coloring to signed graphs. Máčajová, Raspaud and Škoviera introduced one of them and conjectured that in this setting, for signed planar graphs four colors are always enough, generalising thereby The Four Color Theorem. We disprove the conjecture.
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。
A Ramsey-type theorem on deficiency
本文证明了关于图亏格的 Ramsey 型定理，刻画了有界亏格图的所有禁用导出子图，并以此回答了 Fujita 等人于 2006 年提出的相关问题。
Nowhere-zero flows in signed graphs: A survey
我们调查已知的结果有关无处零流和相关的主题，如电路覆盖和结构的电路的符号图。我们包括一个概述了几种不同的定义签署图着色。
Let $S=(G, σ)$ be a signed graph where $G=(V, E)$ is a graph called the underlying graph of $S$ and $σ:E(G) \rightarrow \{+,~-\}$. Let $f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$ such that $σ(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity and $σ(uv)=-$ if and only if $f(u)$ and $f(v)$ are of opposite parity. Under $f$ we get a signed graph $G_f$ denoted as $S$, which is a parity signed graph. In this paper, we initiate the study of parity labeling in signed graphs and we define and find `rna' number denoted as $σ^-(S)$ for some classes of signed graphs. We also characterize some sign…
A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed $K_3$'s, three signed $K_4$'s, and seven signed $K_5$'s. In this paper, we prove that there are sixteen signed $K_6$'s upto switching ismomorphism.
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…
The spectral radius of unbalanced signed graphs without negative $C_3$ or $C_4$
本文研究了固定阶数下不含负三角形和负四边形的非平衡符号图，确定了其中谱半径最大的图的结构。
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 Nullity of Bicyclic Signed Graphs
设Γ是一个符号图，A（Γ）是Γ的邻接矩阵.Γ的零度是A（Γ）的谱中特征值零的重数。本文刻画了零度为n-2或n-3的n阶符号图，并引入了一种保持零度的图变换。作为应用，我们确定了n阶不平衡双圈符号图的零度为n-3或n-4，以及n阶符号双圈符号图（包括简单双圈图）的零度为n-5。
Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs
该研究证明了布切特猜想在循环五边连通立方符号图中成立，表明此类图具有处处非零的六流。
The signed graphs with symmetric spectra
本文研究具有对称谱的符号图的刻画问题，给出了谱对称符号图的必要和（或）充分条件，并提出了构造此类符号图的方法，生成了无限族符号图。
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。
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…
Construction of additively graceful signed graphs-I
本文研究了如何从给定图G构造加性优雅有向图S，并探讨了加性优雅有向图的构造方法及非可除求和图G(m, n)的加性优雅标记值。
Turán-type extremal problems for unbalanced signed graphs
本文推广Turán定理和Nikiforov谱Turán定理至符号图，并确定了无平衡完全符号子图的不平衡符号图的第二最大谱半径。
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