Given a non-abelian finite group G, let π(G) denote the set of prime divisors of the order of G and denote by Z(G) the center of G. The prime graph of G is the graph with vertex set π(G) where two distinct primes p and q are joined by an edge if and only if G contains an element of order pq and the non-commuting graph of G is the graph with the vertex set G−Z(G) where two non-central elements x...

This article is the third in a series of four articles (preceded by [19,20] and continued in [18]) about modelling circuits by many sorted algebras. A circuit is defined as a locally-finite algebra over a circuit-like many sorted signature. For circuits we define notions of input function and of circuit state which are later used (see [18]) to define circuit computations. For circuits over mono...

A graph G is self complementary if it is isomorphic to its complement G. In this paper we define bipartite self-complementary graphs, and show how they can be used to understand the structure of self-complementary graphs. For G a selfcomplementary graph of odd order, we describe a decomposition of G into edge disjoint subgraphs, one of which is a bipartite self-complementary graph of order |G|−...

For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and {i, j} is an edge in G. Building upon recent work involving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corank of a di...

We study packings of graphs with given maximal degree. We shall prove that the (hitherto unproved) Bollobás–Eldridge–Catlin conjecture holds in a considerably stronger form if one of the graphs is d-degenerate for d not too large: if d,∆1,∆2 ≥ 1 and n > max{40∆1 ln ∆2, 40d∆2} then a d-degenerate graph of maximal degree ∆1 and a graph of order n and maximal degree ∆2 pack. We use this result to ...

Much emphasis has been placed in recent years on deriving or calculating programs rather than proving them correct. Adequate calculational frameworks are needed to support such an approach. The present work explores the use of a calculus of relations to express and reason about graph properties in an algorithmic context. We take a generic program that computes a maximal set, over some universe,...

Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.

For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and {i, j} is an edge in G. Building upon recent work involving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corank of a di...

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of A-quotient-free sets when A belongs to a particular class. It is known that in the case A = {p, q}, where p, q are coprime integers greater than one, the lates...

This paper proposes a new algorithm for solving maximal cliques for simple undirected graphs using the theory of prime numbers. A novel approach using prime numbers is used to find cliques and ends with a discussion of the algorithm.