Graphs with prescribed radius, diameter, and center

TL;DR

This paper proves that for any given radius, diameter, and center graph satisfying certain conditions, one can construct a connected graph with those properties, demonstrating a flexible relationship among these parameters.

Contribution

It introduces a method to construct connected graphs with prescribed radius, diameter, and center, expanding understanding of graph parameter realizability.

Findings

Existence of connected graphs with specified radius, diameter, and center for given parameters.

Construction method applicable to any finite simple graph as center.

Parameter constraints: 1<r<d≤2r.

Abstract

Among other things, it is shown that for every pair of positive integers $r$, $d$, satisfying $1<r<d\leq 2r$, and every finite simple graph $H,$ there is a connected graph $G$ with diameter $d$, radius $r$, and center $H.$