Related papers: Locally $s$-distance transitive graphs

In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and…

For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class.…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma(v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and for any field $F$, let…

Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each…

Vertex-stabilizers of trivalent edge-transitive graphs have been classified by Tutte, Goldschmidt and some others in several previous papers. Tetravalent half-arc-transitive graphs form an important class of tetravalent edge-transitive…

A graph $\Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$, and connected-homogeneous (CH) if this…

Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the…

A subgroup of the automorphism group of a graph $\G$ is said to be {\em half-arc-transitive} on $\G$ if its action on $\G$ is transitive on the vertex set of $\G$ and on the edge set of $\G$ but not on the arc set of $\G$. Tetravalent…

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a…

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…

We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is…

A graph is prime if it does not admit a partition $(A,B)$ of its vertex set such that $\min\{|A|,|B|\} \geq 2$ and the rank of the $A\times B$ submatrix of its adjacency matrix is at most $1$. A vertex $v$ of a graph is non-essential if at…

A graph $G$ realizes the degree sequence $S$ if the degrees of its vertices is $S$. Hakimi gave a necessary and sufficient condition to guarantee that there exists a connected multigraph realizing $S$. Taylor later proved that any connected…

In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically…

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In…

Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge connects two…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…

This paper gives a sufficient and necessary condition for the existence of an (X, s)-arc-transitive imprimitive graph which is not a multicover of a given quotient graph.