Related papers: On the diameter of Engel graphs

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the…

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence…

A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the…

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 Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much…

In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5--17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this…

The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that…

The \emph{local edge-length ratio} of a planar straight-line drawing $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$ that share a common vertex. The \emph{global edge-length ratio} of $\Gamma$ is the…

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let $\{G_1, G_2, \ldots,…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$.…

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) +…

Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final…

In this paper, we introduce a new model to study network reliability with node failures. This model, strongly connected node reliability, is the directed variant of node reliability and measures the probability that the operational vertices…

We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse…

We give a combinatorial proof of the following theorem. Let $G$ be any finite group acting transitively on a set of cardinality $n$. If $S \subseteq G$ is a random set of size $k$, with $k \geq (\log n)^{1+\varepsilon}$ for some…

Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $\pi(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow…

For a group $\Gamma$, a $\Gamma$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $\Gamma$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is…

Suppose $\Gamma$ is a finite simple graph. If $D$ is a dominating set of $\Gamma$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\Gamma$, then we say that $D$ is an odd dominating set for $\Gamma$. For a…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $\Gamma_H(G)$ whose vertex set $V(\Gamma_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if…