<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [<xref ref-type="bibr" rid="j_ausm-2023-0019_ref_018">18</xref>] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.</p>
</abstract>

Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.

A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

Department of Mathematics, Faculty of Engineering and Natural Sciences

Department of Mathematics, Faculty of Arts and Science

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the...

A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

Published Online: Dec 26, 2023

Page range: 337 - 353

Received: May 28, 2021

DOI: https://doi.org/10.2478/ausm-2023-0019

Keywords
nullity, maximal nullity condition, minimal nullity condition, omega invariant, matching number

© 2023 Mert Sinan Oz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

Published Online: Dec 26, 2023

Page range: 337 - 353

Received: May 28, 2021

DOI: https://doi.org/10.2478/ausm-2023-0019

Keywordsnullity, maximal nullity condition, minimal nullity condition, omega invariant, matching number

© 2023 Mert Sinan Oz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
nullity, maximal nullity condition, minimal nullity condition, omega invariant, matching number