Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least $-\frac{n^2}{25}$. Also we provide an example of a weighted graph with described properties and the sum of weights $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$.
The previous best known bounds were $-\frac{n^2}{16}$ and $-(1+o(1))\frac{n^2}{54}$ respectively. We show that the constant $-1/54$ is optimal under some additional conditions.
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 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.
The notion of an Abstract Young (briefly: AY) representation is a natural generalization of the classical Young orthogonal form. The AY representations of the symmetric group are characterized by Adin, Brenti and Roichman in [U2]. In this paper we present several types of minimal AY representation of $D_n$ associated with standard D-Young tableaux which are a natural generalization of usual standard Young tableaux. We give an explicit combinatorial view (the representation space is spanned by certain standard tableaux while the action is a generalized Young orthogonal form) of representations …
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…
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…
We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed…
A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free W…
A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper, we show that if $G$ is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then $G$ is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product $G\Box H$ is well-edge-dominated, where $G$ and $H$ have order at least $2$, if and only if $G…
If each minimal dominating set in a graph is a minimum dominating set, then the graph is called well-dominated. Since the seminal paper on well-dominated graphs appeared in 1988, the structure of well-dominated graphs from several restricted classes have been studied. In this paper we give a complete characterization of nontrivial direct products that are well-dominated. We prove that if a strong product is well-dominated, then both of its factors are well-dominated. When one of the factors of a strong product is a complete graph, the other factor being well-dominated is also a sufficient cond…
Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph $G$ is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that recognizing well-dominated graphs can be done polynomially for graphs without cycles of lengths $4$ and $5$, by proving that a graph belonging to this family is well-dominated if and only …
A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order $2$. Under the assumption that at least one of the connected graphs $G$ or $H$ has no isolatable vertices, we prove that the direct product of $G$ and $H$ is well-dominated if and only if either $G=H=K_3$ or $G=K_2$ and $H$…
A class ${\cal F}$ of graphs is called {\em tame} if there exists a constant $k$ so that every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^k)$ minimal separators, {\em strongly-quasi-tame} if every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^{k \log n})$ minimal separators, and {\em feral} if there exists a constant $c > 1$ so that ${\cal F}$ contains $n$-vertex graphs with at least $c^n$ minimal separators for arbitrarily large $n$. The classification of graph classes into tame or feral has numerous algorithmic consequences, and has recently received considerable a…
A graph $G$ is well-covered if every minimal vertex cover of $G$ is minimum, and a graph $G$ is well-dominated if every minimal dominating set of $G$ is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were first introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs was proved coNP-complete by [Chvátal and Slater, AODM 1993] and by [Sankaranarayana and Stewart, Networks 1992], but the complexity of r…
