Related papers: On certain edge-transitive bicirculants

In this note, we construct bipartite 2-walk-regular graphs with exactly 6 distinct eigenvalues as incidence graphs of group-divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We…

A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$…

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified

We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each…

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not…

We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface. Previously, such decompositions had only been known for only a few complete graphs. These…

A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of…

A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a…

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each…

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs…

A Cayley graph $\Cay(G,S)$ is said to be inner-automorphic if $S$ is a union of conjugacy classes of a group $G$, and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four…

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and \v{S}koviera and…

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency…

This paper is to accompany the Census of Edge-Transitive Tetravalent Graphs, available at jan.ucc.nau.edu/~swilson/C4FullSite/index.html, which is a collection of all known edge-transitive graphs of valence 4 up to 512 vertices. The Census…

For a graph $\Gamma$, the {\em distance} $d_\Gamma(u,v)$ between two distinct vertices $u$ and $v$ in $\Gamma$ is defined as the length of the shortest path from $u$ to $v$, and the {\em diameter} $\mathrm{diam}(\Gamma)$ of $\Gamma$ is the…

A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms…

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…

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…

It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite…

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ is 4-valent, connected and $G$-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive…