On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

Abstract

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where p>q are odd primes.

Classifying Seven-Valent Symmetric Graphs of Order 8pq

“…Since Aut(T) = PSL (3,8). Z 6 , we have A ∼ = PSL (3,8).Z 2 , PSL (3,8).Z 3 or PSL (3,8).Z 6 , and thus |A α | = 2 7 • 3 2 • 7, 2 6 • 3 3 • 7 or 2 7 • 3 3 • 7, which is impossible according to Lemma 2. Thus, T is transitive on VΓ.…”

“…Similarly, if r = 2, then T = Sz(8), PSp(4, 8), PSL(2, 2 6 ), PSL(2, 2 9 ), PSL (3,8), PSL (3,16), PSL(4, 4), PSL(5, 2), PSL(6, 2), 3…”

“…Assume that T ∼ = PSL (3,8). If T has two orbits on VΓ, then Γ is bipartite and |T α | = 2 7 • 3 2 • 7.…”

“…Among these parameters, the spectral characterizations, main eigenvalues, and distance characteristic polynomials are the better ones to measure the stability of a network; see [3][4][5][6][7], for example. For arc-transitivity, see [8], as an example. In this paper, we study the arc-transitivity of graphs.…”