Related papers: Power Graph and Exchange Property for Resolving Se…

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some…

The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these…

Let $G$ be a connected graph. A vertex $w$ strongly resolves a pair $u$, $v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a strong resolving…

Exchangeability is a desired statistical property of network ensembles requiring their invariance upon relabelling of the nodes. However combining sparsity of network ensembles with exchangeability is challenging. Here we propose a…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension of $\Gamma$ is the smallest size of…

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have…

A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the…

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and…

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian…

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

The power graph $\mathcal{P}(G)$ is a graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is a graph with vertex set…

Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$…

The power graph $P(G)$ of a group $G$ is a simple graph with the vertex set $G$ such that two distinct vertices $u,v \in G$ are adjacent in $P(G)$ if and only if $u^m = v$ or $v^m = u$, for some $m \in \mathbb{N}$. The purpose of this paper…

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple…

Metacirculants are a rich resource of many families of interesting graphs, and weak metacirculants are generalizations of them. A graph is called a {\em split weak metacirculant} if it has a vertex-transitive split metacyclic automorphism…

It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each…

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the…

Let $R$ be a ring with unity. The clean graph $\text{Cl}(R)$ of a ring $R$ is the simple undirected graph whose vertices are of the form $(e,u)$, where $e$ is an idempotent element and $u$ is a unit of the ring $R$ and two vertices $(e,u)$,…

The power graph of a given finite group is a simple undirected graph whose vertex set is the group itself, and there is an edge between any two distinct vertices if one is a power of the other. In this paper, we find a precise formula to…