Related papers: Groups for which it is easy to detect graphical re…

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…

A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more…

We show that if a group can be represented as a graph product of finite directly indecomposable groups, then this representation is unique.

We prove that the subgroup graph of a finite group $G$ is regular if and only if $G$ is cyclic with square-free order.

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…

The concept of directed strongly regular graphs (DSRG) was introduced by Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several construction…

A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$…

In this paper, the notion of proper proximality (introduced in [BIP18]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that for any two edges, the…

The study of ORR was inspired by L\'{a}zsl\'{o} Babai in 1980 when he asked a question: Which [finite] groups admit an oriented graph as a DRR? And it has been solved by Joy Morris and Pablo Spiga through a series of papers in 2018. In this…

This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the…

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with…

The directed power graph $\mathcal G(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ in which $x\rightarrow y$ if $y$ is a power of $x$, the power graph is the underlying simple graph, and the enhanced power…

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ where $R$ is a "reduction", a…

Given a finite group $G$, its prime graph $\Gamma(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We…

A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $\Delta(G)$ as the (simple undirected) graph whose vertices are the prime…

A Cayley (di)graph $\Cay(G,S)$ of a finite group $G$ is called CI if, for every Cayley (di)graph $\Cay(G,T)$ of $G$, $\Cay(G,S)\cong \Cay(G,T)$ implies that $S^{\sigma}=T$ for some $\sigma\in \Aut(G)$. The group $G$ is called an NDCI-group…

Let ${\rm GK}(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that…

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 $\BCay(G,S)$…