Heptavalent Symmetric Graphs of Order 16p

Abstract

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. We classify connected heptavalent symmetric graphs of order 16p for each prime p. As a result, there are two such sporadic graphs with p = 3 and 7, and an infinite family of 1-regular normal Cayley graphs on the group [Formula: see text] with 7|(p – 1).

Classifying Seven-Valent Symmetric Graphs of Order 8pq

“…More recently, a large number of papers on seven-valent symmetric graphs have been published. The classification of seven-valent symmetric graphs of order 8p, 12p, 16p, 24p or 2pq were presented in [18][19][20][21][22]. We shall generalize these results by determining all connected seven-valent symmetric graphs of the order 8pq.…”

“…If p = 2, then Γ has the order 16q; by [20], we have q = 3, 7 or 7 q − 1, and Γ is isomorphic to C 48 , C 112 or C (2 3 ,2q) . If p = 3, then Γ has the order 24q; in [21], we have q = 5 or 13, and Γ is isomorphic to C 120 , C i 312 with i = 1, 2, 3, 4, C 5 312 or C 6 312 .…”

Valency seven symmetric graphs of order $2pq$

“…Recently, Guo et al [15] gave the structure of vertex stabilizers of valency seven symmetric graphs, and this encourages us to consider some work on valency seven symmetric graphs. In [16], Guo et al classified valency seven symmetric graphs of order 4p, and in [25], Pan et al classified primevalent symmetric graphs of square-free order. But, we obtain this result for valency seven symmetric graphs of order 2pq independently.…”

Prime-valent arc-transitive basic graphs with order 4 p or 4 p 2

“…In the literature, the classification of arc-transitive graphs of small valency has been extensively studied; refer to [4-7, 9, 10, 15, 19-23, 27-29] and references therein. In particular, cubic and pentavalent arc-transitive graphs of order 4p or 4p 2 are classified in [13,16], and heptavalent arc-transitive graphs of order 4p are classified in [8], where p is a prime. The purpose of this paper is to characterize prime-valent arc-transitive basic graphs with order four times a prime or a prime square.…”