Related papers: Two-connected signed graphs with maximum nullity a…

A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…

Let $\Sigma=(\Gamma, \sigma)$ is a signed graph(or sigraph in short), where $\Gamma$ is a underlying graph of $\Sigma$ and $\sigma:E\longrightarrow \{+, -\}$ is a function. Consider $\Gamma=Cay(\mathbb{Z}_{p_{1}}\times…

Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$. One can also consider the same graph…

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a…

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…

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 $T$.…

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained…

A signed graph (SG) is a graph where edges carry sign information attached to it. The sign of a network can be positive, negative, or neutral. A signed network is ubiquitous in a real-world network like social networks, citation networks,…

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $[n]=\{1,2, \ldots ,n\}$ is the graph on $[n]$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. Let $H$ be a fixed graph…

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…

A seminal result by Whitney describes when two graphs have the same cycles. We consider the analogous problem for even cycle matroids. A representation of an even cycle matroid is a pair formed by a graph together with a special set of…

A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency…

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by…

A graph is equimatchable if each of its matchings is a subset of a maximum matching. It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint. This paper provides…

For distinct vertices $u,v$ in a graph $G$, let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $\kappa_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every…

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…

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…

Let $\mathcal{F}$ be a set of connected graphs, and let $G$ be a graph. We say that $G$ is \emph{$\mathcal{F}$-free} if it does not contain $F$ as an induced subgraph for all $F\in\mathcal{F}$, and we call $\mathcal{F}$ a forbidden pair if…