Related papers: On CIS Circulants
A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\in\Psi(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$,…
The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral…
Given a hypergraph $H$, the dual hypergraph of $H$ is the hypergraph of all minimal transversals of $H$. A hypergraph is conformal if it is the family of maximal cliques of a graph. In a recent work, Boros, Gurvich, Milani\v{c}, and Uno…
Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. A bi-Cayley graph $\BCay(G,S)$…
A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is…
A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover…
The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…
A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…
The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that…
A circulant graph is a simple graph whose adjacency matrix can be represented in the form of a circulant matrix, while a nut graph is considered to be a graph whose null space is spanned by a single full vector. In a previous study by…
A Cayley graph on a group $G$ has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of the normaliser of the regular representation of $G$. A group…
Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned…
A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph $G$, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope…
The smallest number of cliques, covering all edges of a graph $ G $, is called the (edge) clique cover number of $ G $ and is denoted by $ cc(G) $. It is an easy observation that for every line graph $ G $ with $ n $ vertices, $cc(G)\leq n…
Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of…
Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…
The cyclic graph $\Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such…
The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that…
The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with…
An independent set $I$ in a graph $G$ is maximal if $I$ is not properly contained in any other independent set of $G$. The study of maximal independent sets (MIS's) in various graphs is well-established, often focusing upon enumeration of…