Maximum edges possible in a graph for restricted independence number, maximum degree, and maximum matching size

TL;DR

This paper establishes precise upper bounds on the number of edges in simple graphs based on restrictions on maximum matching size, independence number, and maximum degree, and constructs extremal graphs achieving these bounds.

Contribution

It provides sharp bounds and constructs extremal graphs for the maximum edges problem under combined parameter restrictions, including uniqueness results.

Findings

Derived tight upper bounds for edge counts

Constructed extremal graphs achieving bounds

Proved uniqueness of extremal graphs when applicable

Abstract

This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct extremal graphs that achieve the edge bounds in all cases. We further establish uniqueness of these extremal graphs whenever they are unique.