Title:Omega Invariant of Complement Graphs and Nordhaus-gaddum Type
Results
Volume: 21
Issue: 3
Author(s): Aysun Yurttas Gunes*
Affiliation:
- Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Gorukle Campus, 16059, Bursa, Turkey
Keywords:
Omega invariant, cyclomatic number, triangular number, complement of a graph, graph parameter, component.
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. Also we used
triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G
and G.
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 and triangular numbers.
Conclusion: Some relations between the omega invariants of a graph and its complement are obtained.
