Related papers: Edge-transitive core-free Nest graphs

Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $\Gamma_{\mathrm{Hom}}(M)$ as the simple graph whose vertices are the proper submodules of $M$, with an edge between distinct vertices $N_1$ and $N_2$ if and…

We prove that if $\Gamma$ is a finite connected vertex-transitive cubic graph, then either $|V\Gamma| \le 90$, or $\Gamma$ is a split Praeger--Xu graph, or there exist two vertices $\alpha$ and $\beta$ such that the identity is the only…

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…

Let $\Gamma$ be a graph with diameter at least two. Then $\Gamma$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of…

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such…

Let $\Gamma$ be an undirected and simple graph. A set $ S $ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components each containing a cycle. If $\Gamma$ has a…

Let $\Gamma$ be a finite X-symmetric graph with a nontrivial X-invariant partition $\mathcal {B}$ on $V(\Gamma)$ such that $\Gamma_{\mathcal {B}}$ is a connected (X,2)-arc-transitive graph and $\Gamma$ is not a multicover of…

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and…

A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy…

A nut graph is a simple graph whose adjacency matrix has the eigenvalue~0 with multiplicity~1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al.~[\emph{Disc. Math. Graph Theory} 40 (2020),…

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph…

Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and…

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two…

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a connected…

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(\Gamma,\Sigma)$ is stable if $Aut(\Gamma\times\Sigma) \cong…

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 nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for…

We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the…

A graph is said to be {\em half-arc-transitive} if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called…

The prime graph of a finite group $G$ is the labelled graph $\Gamma(G)$ with vertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A group $G$ is recognisable by its prime graph…