Omega invariant of complement graphs and nordhaus-gaddum type results

Abstract

Aims: To obtain relations between the omega invariants of a graph and its complement. Background: We aim to use some graph parameters including the cyclomatic numbers, number of components, maximum number of components, order and size of both graphs G and (G) over bar. Also we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of.. and (G) over bar.Objective: Several bounds for the above graph parameters will be given by direct application of omega invariant.Methods: We use combinatorial and graph theoretical methods to study formulae, relations and bounds on the omega invariant, the number of faces and the number of components of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type results in our calculations. In these calculations, the number of triangular numbers less than a given number plays an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized.Result: In this paper, we obtained relations between the omega invariants of a graph and its complement in terms of several graph parameters such as the cyclomatic numbers, number of components, maximum number of components, order and size of G and (G) over bar and triangular numbers.Conclusion: Some relations between the omega invariants of a graph and its complement are obtained.