Paper Info
Title
A New Family of Algebraically Defined Graphs with Small Automorphism Group
Abstract
Let p be an odd prime, q = p(e), e >= 1, and F = F-q denote the finite field of q elements. Let f : F-2 -> F and g : F-3 -> F be functions, and let P and L be two copies of the 3-dimensional vector space F-3. Consider a bipartite graph Gamma(F) (f,g) with vertex partitions P and L and with edges defined as follows: for every (p) = (P-1, P-2, P-3)is an element of P and every [l] = [l(1),l(2),l(3)]is an element of L, {(p), [l]} = (p) [l] is an edge in Gamma(F) (f,g) if P-2 + l(2) = f (P-1, l(1)) and p(3) + l(3) = g(p(1), p(2), l(1)). The following question appeared in Nassau: Given Gamma(F) (f,g), is it always possible to find a function h : F-2 -> F such that the graph Gamma(F) (f, h) with the same vertex set as Gamma(F)(f,g) and with edges (p)[l] defined in a similar way by the system P-2 + l(2) = f (P-1, l(1)) and p(3) + l(3) = h(p(1), l(1)), is isomorphic to Gamma(F) (f, g) for infinitely many q? In this paper we show that the answer to the question is negative and the graphs Gamma(Fp) (p1l1, p(1)l(1)p(2)(p(1) + P-2 + P1P2)) provide such an example for p (math) 1 (mod 3). Our argument is based on proving that the automorphism group of these graphs has order p, which is the smallest possible order of the automorphism group of graphs of the form Gamma(F)(f, g).
Year
DOI
Venue
2022
10.37236/10707
ELECTRONIC JOURNAL OF COMBINATORICS
DocType
Volume
Issue
Journal
29
1
ISSN
Citations 
PageRank 
1077-8926
0
0.34
References 
Authors
0
2
Name
Order
Citations
PageRank
Felix Lazebnik135349.26
Vladislav Taranchuk200.34