Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. Although the basic properties of these polynomials are known to some extent, there is, to my knowledge, no detailed development available. My aim in writing this article is to fill in this gap. Notation In the following we will use capital letters for (algebraic) polynomials, fo...

In this note we consider two related infinite families of graphs, which generalize the Petersen and the Coxeter graph. The main result proves that these graphs are cores. It is determined which of these graphs are vertex/edge/arc-transitive or distance-regular. Girths and odd girths are computed. A problem on hamiltonicity is posed.

Double orthogonality in the set of eigenvectors of any symmetric graph matrix is exploited to propose a set of nodal centrality metrics, that is “ideal” in the sense of being complete, uncorrelated and mathematically precisely defined and computable. Moreover, we show that, for each node m, such a nodal eigenvector centrality metric reflects the impact of the removal of node m from the graph at...

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arctransitive regular cyclic covers of the complete bipartite graph Kp,p for each odd prime p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime p. The regular elementary a...

A regular graph is called semisymmetric if it is edge transitive but not vertex transitive It is proved that the Gray graph is the only cubic semisymmetric graph of order p where p is a prime

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with 1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes. The classification of finite simple groups i...

We construct continuum many non-isomorphic countable digraphs which are highly arc-transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive. 2000 Mathematics Subject Classification: 20B07, 20B15, 03C50, 05C20.

Random structures often present a trade-off between realism and tractability, the latter predominantly enabled by independence. In pioneering random graph theory, Erdős and Rényi originally studied the set of all graphs with a given number of edges, seeking to identify its typical properties under the uniform measure, i.e., the G(n,m) model of random graphs. However, this approach quickly prese...

Introduction-The question whether an isomorphism test for two graphs may be found, which is polynomial in the number of vertices, , n stands open for quite a while now. The purpose of the present article is to answer this question affirmatively by presenting an algorithm which decides whether two graphs are isomorphic or not and showing that the number of n independent elementary operations nee...

A regular and edge transitive graph which is not vertex transitive is said to be semisymmetric Every semisymmetric graph is necessarily bipartite with the two parts having equal size and the automorphism group acting transitively on each of these two parts A semisymmet ric graph is called biprimitive if its automorphism group acts prim itively on each part In this paper a classi cation of bipri...