Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

Abstract

A regular cover of a connected graph is called cyclic or dihedral if its transformation group is cyclic or dihedral respectively, and arc-transitive (or symmetric) if the fibrepreserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.

Polynomials of degree 4 over finite fields representing quadratic residues

“…Let d(n, m) and c 4 (n, m) be the corresponding values for I q as defined by functions d and c r in (2.1) and (2.2). Then one can easily see that d(3, 6) ≤ d (3,6) and that c 4 (3, 6) ≥ c 4 (3, 6), which implies d(3, 6) − c 4 (3, 6) ≥ d(3, 6) − c 4 (3,6) . Therefore, (2.3) holds for J q if it holds for I q .…”

“…The motivation for this paper is twofold: first refining the result of Madden and Vélez about polynomials that represent quadratic residues at primitive roots [9], and in doing so obtaining a tool with which hamiltonicity of certain families of vertex-transitive graphs of order a product of two primes is proved via a structural analysis of their quotients with respect to an automorphism of prime order. Such a connection between algebraic graph theory and finite fields is not surprising, see, for example, [6,14] for a similar application of finite fields.…”

“…then there exists a primitive root β of F such that f (β) = β 4 + kβ 2 + 1 is a square in F except when (p, k) ∈ {(5, 4), (13, 1), (13,4), (13,5), (13,6), (13,7), (13, 10), (37, 3), (37, 28), (37, 29), (61, 18), (61, 37), (61, 40)}.…”

Generalised Voltage Graphs

“…At the other extreme, if a group of automorphisms G acts semiregularly on the vertices of Γ (that is, G v = 1 for every vertex v ∈ V(Γ)) one can reconstruct Γ as the derived covering graph Cov(Γ/G, ζ), where Γ/G is the quotient graph and ζ : D(Γ) → G is a mapping called a voltage assignment. The theory of graph covers and their description in terms of voltages has a long history going back to the work of Gross and Tucker [14,15] and has now become one of the central tools in the theory of symmetries of graphs (see [6,8,9,16,18,19,25], for exmaple).…”