Tetravalent one-regular graphs of order 4p2

Abstract

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

An Algorithmic-type Classification of Tetravalent One-regular Graphs Using Computer Algebra Tools

“…Also, by [6,35,36,44,50, 51] every tetravalent one-regular graph of order pq or p 2 is a circulant graph. Furthermore, the classification of tetravalent one-regular graphs of order 3p 2 , 4p 2 , 5p 2 , 6p 2 and 2pq are given in [8,18,19,20,54]. Along this line we obtain Theorem 3.1, which is the main result of the paper.…”

Two-arc-transitive two-valent digraphs of certain orders

“…It is known that there are precisely eight non-abelian simple groups whose orders are divisible by at most three distinct primes (see, for example, [12]); these are Alt(5), PSL(2, 7), Alt(6), PSL (2,8), PSL (2,17), PSL (3,3), PSU (3,3), and PSU (4,2). Out of these, only the first five are such that the odd primes appear with multiplicity at most 2; these five groups, together with their orders and the orders of their automorphism groups are listed in Table 2.…”

“…If p = 7, then the order 2 a 3 b 7 c of Γ is larger than 1000 only when a = 3, b = 1 and c = 2. Since 9 divides the order of PSL (2,8), this implies that T ∼ = PSL (2,8), and therefore T ∼ = PSL(2, 7) and |V (Γ)| = 8 • 3 • 7 2 = 1176.…”

“…Once the prime factorisation of the order becomes more complex, results of this type become considerably more complicated (see [38] for an illustration of the difficulties that can arise when the order is a product of three distinct primes). However, when one fixes the valence (and perhaps imposes some further restrictions), further analysis becomes possible (see for example [5,8,10]).…”

A family of tetravalent one-regular graphs

“…Also, by [5,27,28,30,34,35] every 4-valent one-regular graph of order pq or p 2 is a circulant graph. Furthermore, the classification of 4-valent oneregular graphs of order 3p 2 , 4p 2 , 6p 2 and 2pq are given in [8,15,17,37]. Along this line the aim of this paper is to classify 4-valent one-regular graphs of order 5p 2 , see Theorem 3.3.…”