A Characterization of Certain Families of 4-Valent Symmetric Graphs

Tetravalent one-regular graphs of order 4p2

“…For a normal subgroup N of G, the quotient graph X N of X relative to the set of orbits of N is defined as the graph whose vertices are orbits of N on V(X) with two orbits being adjacent in X N if there is an edge between these two orbits in X. The following proposition is a 'reduction' theorem which is deduced from To state the next result we need to introduce three families of tetravalent graphs that were first defined in [23]. First, let C ±1 (p; 4, 2) be the graph with vertex set Z …”

“…To state the next result we need to introduce two additional families of tetravalent graphs that were first defined in [23]. The graph C ±1 (p; 4p, 1) is defined to have the vertex set Z p × Z 4p and the edge set…”

“…The aim of this paper is to classify tetravalent one-regular graphs of order 4p 2 , see Theorem 5.1. (For more results on tetravalent arc-transitive graphs, see [22,23,27,33]. )…”

Tetravalent one-regular graphs of order 2pq

“…Let X be a tetravalent connected symmetric graph, and N an elementary Abelian p-group. A classification of tetravalent connected symmetric graphs was obtained for the case when N has at most two orbits in [11], and a characterization of such graphs was given for the case when X N is a cycle in [12]. The following is a 'reduction' theorem.…”

“…It follows from the classification that with the exception of four graphs of orders 12 and 30, all such graphs are Cayley graphs on Abelian, or dihedral, or generalized dihedral groups. For more results on tetravalent symmetric graphs, see [11,12,17,28].…”