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ISSN (Print): 1570-1794
ISSN (Online): 1875-6271

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Research Article

Omega Invariant of Complement Graphs and Nordhaus-gaddum Type Results

Author(s): Aysun Yurttas Gunes*

Volume 21, Issue 3, 2024

Published on: 05 October, 2023

Page: [298 - 302] Pages: 5

DOI: 10.2174/1570179421666230914151600

Price: $65

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.

Keywords: Omega invariant, cyclomatic number, triangular number, complement of a graph, graph parameter, component.

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Graphical Abstract

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