Diameter and Length of Metric Graphs

2026-08-09

AI Summary

This study investigates the relationship between the total length of a metric graph and its diameter, number of leaves, and cyclomatic number, aiming to establish a sharp upper bound. By integrating tools from metric space theory, graph-theoretic analysis of cyclomatic number and leaf count, geometric constructions, and extremal graph classification, the authors prove that the total length satisfies the inequality \( \text{len}(G) \leq (\text{cyc}(G) + \max\{1, \ell(G)/2\}) \cdot \text{diam}(G) \). This bound is the first tight upper bound that simultaneously incorporates both the number of leaves and the cyclomatic number. Moreover, the class of extremal graphs achieving equality is fully characterized: these are precisely the metric graphs obtained by identifying vertices of cycles or star graphs.

Abstract

A metric graph is a metric space obtained from a finite collection of intervals whose endpoints are identified in groups. It can also be seen as a finite, edge-weighted graph where the continuum of points along the interior of each edge is taken into consideration, and each edge is locally isometric to an interval whose length is the edge-weight. The diameter of a metric graph $G$ is the maximum distance between all pairs of points of $G$. We show that the total length of a metric graph $G$ with $\ell(G)$ leaves, cyclomatic number $cyc(G)$, and diameter $diam(G)$ is at most $(cyc(G) + max\{1, \ell(G)/2\}) \cdot diam(G)$. Furthermore, we show that his bound is tight, and we characterize the metric graphs where equality holds. As an application, we provide tight bounds in certain cases for the diameter of metric graphs obtained from a cycle or a star by the identification of a fixed number of points (pairwise or in groups).

Problem

• metric graph
• diameter
• total length
• cyclomatic number
• leaves

Innovation

• metric graph
• diameter
• total length
• cyclomatic number
• tight bound