Related papers: Graph product and the stability of circulant graph…
Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining…
Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the…
The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness,…
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…
If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…
A {\it graph product} $G$ on a graph $\Gamma$ is a group defined as follows: For each vertex $v$ of $\Gamma$ there is a corresponding non-trivial group $G_v$. The group $G$ is the quotient of the free product of the $G_v$ by the commutation…
Understanding which system structure can sustain stable dynamics is a fundamental step in the design and analysis of large scale dynamical systems. Towards this goal, we investigate here the structural stability of systems with a random…
The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While…
We define graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products.
The goal of the paper "A new graph over semi-direct products of groups" is to define a graph \Gamma(G) on a group G when G splits over a normal subgroup. We demonstrate herein that the graph is ill-defined. We also attempt to ascertain…
We define a family of graphs we call dual systolic graphs. This definition comes from graphs that are duals of systolic simplicial complexes. Our main result is a sharp (up to constants) isoperimetric inequality for dual systolic graphs.…
Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal…
We consider several graphs classes defined in terms of conditions on cliques and stable sets, including CIS, split, equistable, and other related classes. We pursue a systematic study of the relations between them. As part of this study, we…
Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…
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…
The cyclic subgroup graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with cyclic subgroups as a vertex set and two distinct vertices $H_1$ and $H_2$ are adjacent if and only if $H_1 \leq H_2$ and there does not exist any…
Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight $1$. In $1994$, Mahadev et al.~introduced a subclass of equistable graphs,…
In this article we discuss cohomological obstructions to two kinds of group stability. In the first part, we show that residually finite groups $\Gamma$ which arise as fundamental groups of compact Riemannian manifolds with strictly…
Let $\Gamma$ be a finite connected $G$-vertex-transitive graph and let $v$ be a vertex of $\Gamma$. If the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$ is permutation isomorphic to…
The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G…