On bounds for some graph invariants

TL;DR

This paper establishes bounds relating the stability and covering numbers of graphs without isolated vertices, introduces the $ ext{alpha}_v$-cover concept, and provides a lower bound on the number of edges based on these invariants.

Contribution

It proves a new inequality linking the stability number, covering number, and $ ext{alpha}_v$-cover, and derives a lower bound on edges involving these parameters and connected components.

Findings

Proved that $ ext{alpha}(G) \\leq au(G)[1+ ext{alpha}(G)- ext{alpha}_v(G)]$.

Established a lower bound on the number of edges based on stability, covering numbers, and connected components.

Discussed conjectures related to the main inequality.

Abstract

Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\alpha(G)\leq \tau(G)[1+\alpha(G)-\alpha_{v}(G)]$. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability number $\alpha(G)$, the covering number $\tau(G)$ and the number of connected components $c(G)$ of $G$. Namely, let $\alpha$ and $\tau$ be two natural numbers and let $$ \Gamma(\alpha,\tau)= \min{\sum_{i=1}^{\alpha}\bin{z_i}{2} | z_1+...+z_{\alpha}= \alpha+\tau {and} z_i \geq 0 \forall i=1,..., \alpha}. $$ Then if $G$ is any graph, we have: $$ |E(G)| \geq \alpha(G)-c(G)+ \Gamma(\alpha(G), \tau(G)). $$