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 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…
Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. The adjacency matrix of $Γ=(G, σ)$ is a square matrix $A(Γ)=A(G, σ)=\left(a_{i j}^σ\right)$, where $a_{i j}^σ=σ\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},σ)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p=\{u_1,u_2,\ldots,u_p\}$ and $V_q=\{v_1,v_2,\ldots,v_q\}$. Let $(K_{p, q},σ)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, …
In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.
A signed graph $(G, Σ)$ is a graph $G$ and a subset $Σ$ of its edges which corresponds to an assignment of signs to the edges: edges in $Σ$ are negative while edges not in $Σ$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,Σ)$ to $(H,Π)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipar…
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…
The set of distinct eigenvalues of a signed digraph $S$ together with their multiplicities is called its spectrum. The energy of a signed digraph $S$ with eigenvalues $z_1,z_2,\cdots,z_n$ is defined as $E(S)=\sum_{j=1}^{n}|\Re z_j|$, where $\Re z_j $ denotes real part of complex number $z_j$. In this paper, we show that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle of length $\equiv 0\pmod 4$ negative and each cycle of length $\equiv 2\pmod 4$ positive is of the form \\ $$ϕ_S(z)=z^n+\sum\limits_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}(-1)^j c_{2j}(S)z^{n-2…
A signed bipartite (simple) graph $(G, σ)$ is said to be $C_{-4}$-critical if it admits no homomorphism to $C_{-4}$ (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to $C_{-4}$. In particular, the 4-color theorem is equivalent to: Given a planar graph $G$, the signed bipartite graph obtained from $G$ by replacing each edge with a negative path of length 2 maps to $C_{-4}$. We prove that, except for one particular sign…
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).
The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a part of the HOMFLY polynomial of a naturally associated link. We also establish some other, more basic properties of this new notion. This leads to new identities involving the original interior polynomial.
We prove inequalities between the densities of various bipartite subgraphs in signed graphs and graphons. One of the main inequalities is that the density of any bipartite graph with girth r cannot exceed the density of the r-cycle. This study is motivated by Sidorenko's conjecture, which states that the density of a bipartite graph F with m edges in any graph G is at least the m-th power of the edge density of G. Another way of stating this is that the graph G with given edge density minimizing the number of copies of F is, asymptotically, a random graph. We prove that this is true locally, i…
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We construct some families of bipartite signed graphs with only two distinct eigenvalues. This leads to constructing infinite families of regular bipartite Ramanujan 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 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…
We complete the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$. The unsigned graphs and the disconnected, the bipartite and the complete signed graphs with this property have already been determined in two earlier papers. Here we deal with the remaining cases.
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 present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.
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.
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…
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…
In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.
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.
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" is a graph $Γ$ where the edges are assigned sign labels, either "$+$" or "$-$". The sign of a cycle is the product of the signs of its edges. Let $\mathrm{SpecC}(Γ)$ denote the list of lengths of cycles in $Γ$. We equip each signed graph with a vector whose entries are the numbers of negative $k$-cycles for $k\in\mathrm{SpecC}(Γ)$. These vectors generate a subspace of $\mathbb R^{\mathrm{SpecC}(Γ)}$. Using matchings with a strong permutability property, we provide lower bounds on the dimension of this space; in particular, we show for complete graphs, complete bipartite graphs…
We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.
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…
Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.
The aim of this work is to investigate the nonnegative signed domination number $γ^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $γ^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Turán we bound $γ^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. …
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 …
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.
In this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices and we describe some of their properties. Notably, we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix. The main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound.
In 2009, Kong, Wang, and Lee introduced the problem of finding the edge-balanced index sets ($EBI$) of complete bipartite graphs $K_{m,n}$, where they examined the cases $n=1$, $2$, $3$, $4$, $5$ and the case $m=n$. Since then the problem of finding $EBI(K_{m,n})$, where $m \geq n$, has been completely resolved for the $m,n=$ odd, odd and odd, even cases. In this paper we find the edge-balanced index sets for complete bipartite graphs where both parts have even cardinality.
The presented paper studies the flow number $F(G,σ)$ of flow-admissible signed graphs $(G,σ)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,σ)$ there is a set ${\cal G}(G,σ)$ of cubic graphs such that $F(G, σ) \leq \min \{F(H,σ_H) : (H,σ_H) \in {\cal G}(G)\}$. We prove that $F(G,σ) \leq 6$ if $(G,σ)$ contains a bridge and $F(G,σ) \leq 7$ in general. We prove better bounds, if there is an element $(H,σ_H)$ of ${\cal G}(G,σ)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,σ) \leq 4$ and the b…
In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets of complete bipartite graphs $K_{m,n}$ by solving the cases where $n=1$, $2$, $3$, $4$, and $5$, and also the case where $m=n$. In an article soon to be published, Krop, Minion, Patel, and Raridan concluded the edge-balanced index set problem for complete bipartite graphs with both parts of odd cardinality. In this paper, we conclude the problem for complete bipartite graphs where the larger part is of odd cardinality and the smaller is of even cardinality.
For a graph $G = (V, E)$, the $γ$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $γ$-sets of $G$, and two $γ$-sets are adjacent if they differ by a single vertex and the two different vertices are adjacent in $G$. An open question in $γ$-graphs is whether every bipartite graph is the $γ$-graph of some bipartite graph. We answer this question in the negative by demonstrating that $K_{2, 3}$ is not the $γ$-graph of any bipartite graph.
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.
The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture when $\mid A\mid $ = 1, 2, 3, or when A is a geometric or arithmetic progression.
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…
A signed graph is a pair $(G, σ)$, where $G$ is a graph and $σ: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, σ)$ a circular $r$-coloring of $(G, σ)$ is an assignment $ψ$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $σ(e)=+$, then $ψ(u)$ and $ψ(v)$ have distance at least $1$, and if $σ(e)=-$, then $ψ(v)$ and the antipodal of $ψ(u)$ have distan…
Recently, Gross, Mansour and Tucker introduced the partial-dual genus polynomial of a ribbon graph as a generating function that enumerates the partial duals of the ribbon graph by genus. It is analogous to the extensively-studied polynomial in topological graph theory that enumerates by genus all embeddings of a given graph. To investigate the partial-dual genus polynomial one only needs to focus on bouquets, i.e. ribbon graphs with only one vertex. In this paper, we shall further show that the partial-dual genus polynomial of a bouquet essentially depends on the signed intersection graph of …
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.
