On some extremal problems in graph theory

TL;DR

This paper investigates extremal properties of various graph invariants, both in weighted and unweighted graphs, focusing on regular graphs and their relationships, bridging combinatorial and analytical perspectives.

Contribution

It introduces new results on extremal graphs for invariants like girth, diameter, and eigenvalues, including their weighted analogs and relationships among extremal graphs.

Findings

Identified extremal unweighted regular graphs for key invariants

Analyzed relationships between extremal graphs for different invariants

Extended invariants to weighted graphs, linking combinatorics and analysis

Abstract

In this paper we are concerned with various graph invariants (girth, diameter, expansion constants, eigenvalues of the Laplacian, tree number) and their analogs for weighted graphs -- weighing the graph changes a combinatorial problem to one in analysis. We study both weighted and unweighted graphs which are extremal for these invariants. In the unweighted case we concentrate on finding extrema among all (usually) regular graphs with the same number of vertices; we also study the relationships between such graphs.