New Constructions of Self-Complementary Cayley Graphs

Abstract

Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products †

“…This means that there exists k 0 ∈ N such that the strong power G ⊠ k is non-Ramanujan for all k ≥ k 0 . We next obtain an explicit value of such k 0 , which is not necessarily the smallest one, proving that such a valid value for k 0 is given by (63). To that end, based on the above explanation, one needs to deal with the inequality…”

“…For n = 1 and n = 5, there exist graphs of order n that are self-complementary and vertex-transitive; they are, respectively, given by K 1 and C 5 . Graphs that are self-complementary and vertex-transitive, and approaches for their construction, received attention in the literature (see, e.g., [6,69,[83][84][85][86][87]). Remark 11.…”

“…The third value ( 9) is attained by the sequence {g ℓ (G)} twenty-seven times. By the proof of Item (b) in Corollary 3, the multiplicity of the third value ( 27) is equal to the absolute value of the difference between the multiplicities of the second-largest and smallest eigenvalues of the graph G. The spectrum of the graph G is given by 36 1 6 36 (−4) 63 (this can be verified by (10) and ( 11)), and the above difference (in absolute value) is indeed equal to |63 − 36| = 27. Next, let G be the 5-cycle graph C 5 , which is a strongly regular graph with parameters srg(5, 2, 0, 1).…”

“…Proposition 3 shows that all strong powers C ⊠ k 5 , with k ≥ 5, are non-Ramanujan graphs. An expression for k 0 , as it is given in (63), was derived in order to reduce the minimal analytical value of k 0 for which the k-fold strong power of C 5 is asserted to be non-Ramanujan for all k ≥ k 0 . It started with an initial value of k 0 = 8 for n = 5, with a more simple initial expression for k 0 , and it was reduced to k 0 = 5 with the closed-form 21 expression in (63).…”

“…An expression for k 0 , as it is given in (63), was derived in order to reduce the minimal analytical value of k 0 for which the k-fold strong power of C 5 is asserted to be non-Ramanujan for all k ≥ k 0 . It started with an initial value of k 0 = 8 for n = 5, with a more simple initial expression for k 0 , and it was reduced to k 0 = 5 with the closed-form 21 expression in (63). Proposition 3, and the numerical experimentation as above, gives that C ⊠ k 5 is a Ramanujan graph if and only if k = 1 or k = 2.…”