Self-Complementary Vertex-Transitive Graphs of Order a Product of Two Primes

Li, Cai Heng; Rao, G. Venkata

doi:10.1017/s0004972713000427

Cited by 6 publications

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“…For the self-complementary vertex-transitive graphs of order a product of two primes, we give a complete classification of these graphs: they are either a lexicographic product of two self-complementary vertex-transitive graphs of prime order or a normal Cayley graph of an abelian group. This result has been published in [4].…”

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Self-Complementary Vertex-Transitive Graphs

Rao

2016

Bull. Aust. Math. Soc.

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A graph is self-complementary if its complement is isomorphic to the graph itself. A graph is vertex-transitive if its full automorphism group is transitive on its vertex set. This dissertation is intended to present our research results on self-complementary vertex-transitive graphs. In particular, we studied the following problems: constructions of self-complementary vertex-transitive graphs, self-complementary vertex-transitive graphs of order a product of two primes, selfcomplementary metacirculants and self-complementary vertex-transitive graphs of prime-cubed order. The main analysis on these problems relies on two pivotal results due to Guralnick et al.[1] and Li and Praeger [2], which characterise the full automorphism group of a self-complementary vertex-transitive graph in the primitive and the imprimitive cases, respectively.For constructions of self-complementary vertex-transitive graphs, there are generally three known methods: construction by partitioning the complementing isomorphism orbits, construction using the coset graphs and the lexicographic product. In this dissertation we developed various alternative construction methods. As a result, we find a family of self-complementary Cayley graphs of non-nilpotent groups and a new construction for self-complementary metacirculants of p-groups.A complementing isomorphism of a self-complementary graph is an isomorphism between the graph and its complement. For the self-complementary vertex-transitive graphs whose automorphism groups are of affine type, we have obtained a characterisation of all their complementing isomorphisms. Furthermore, we provide a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. This is the first known example with this property in the literature.

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confidence: 64%

Self-Complementary Vertex-Transitive Graphs

Rao

2016

Bull. Aust. Math. Soc.

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“…Since 2000, the study of self-complementary vertex-transitive graphs has been advanced by the work in [10,13]. The self-complementary vertex-transitive graphs whose orders are products of two primes are classified by the authors in [15].…”

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On finite self-complementary metacirculants

Rao

Song

2014

J Algebr Comb

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We prove that the automorphism group of a self-complementary metacirculant is either soluble or has A 5 as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.

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“…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.…”

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

Sason

2023

Entropy

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This paper provides new observations on the Lovász θ-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász θ-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász θ-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász θ-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

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Self-Complementary Vertex-Transitive Graphs of Order a Product of Two Primes

Abstract: In this short paper, we characterise graphs of order pq with p, q prime which are self-complementary and vertex-transitive.2010 Mathematics subject classification: primary 05C25; secondary 05E18.

Cited by 6 publications

References 17 publications

Self-Complementary Vertex-Transitive Graphs

Self-Complementary Vertex-Transitive Graphs

On finite self-complementary metacirculants

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

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