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Abundancy of $z$-\v Soltés' digraphs

Stijn Cambie

TL;DR

This work addresses the existence and abundance of $z$-Šoltés' digraphs, digraphs for which removing any vertex changes the total distance by exactly $z$, with $W(D)$ denoting the Wiener index. It develops a constructive framework based on circulant digraphs $D(n,S)$ to realize any integer difference $z$ by careful choice of $n$ and $S$ and a detailed analysis of vertex transmissions and detours. The authors prove there are infinitely many negative-$z$ and infinitely many Šoltés' digraphs, including explicit examples such as $D(11,\{1\}) \cong C_{11}$ and $D(85,\{4\})$, plus non-vertex-transitive instances and a $3306$-vertex digraph with trivial automorphism group, signaling strong abundance. These results support the graph-case intuition that abundance of negative-Šoltés' structures drives abundance of Šoltés' structures in the digraph setting, and they provide a rigorous digraph analogue to related conjectures and questions. The paper also includes detailed computations and appendices detailing the construction and its properties.

Abstract

We prove the existence of infinitely many \v Soltés' digraphs, the digraph analogue of \v Soltés' graphs. We also give an example of a \v Soltés' digraph with trivial automorphism group.

Abundancy of $z$-\v Soltés' digraphs

TL;DR

This work addresses the existence and abundance of -Šoltés' digraphs, digraphs for which removing any vertex changes the total distance by exactly , with denoting the Wiener index. It develops a constructive framework based on circulant digraphs to realize any integer difference by careful choice of and and a detailed analysis of vertex transmissions and detours. The authors prove there are infinitely many negative- and infinitely many Šoltés' digraphs, including explicit examples such as and , plus non-vertex-transitive instances and a -vertex digraph with trivial automorphism group, signaling strong abundance. These results support the graph-case intuition that abundance of negative-Šoltés' structures drives abundance of Šoltés' structures in the digraph setting, and they provide a rigorous digraph analogue to related conjectures and questions. The paper also includes detailed computations and appendices detailing the construction and its properties.

Abstract

We prove the existence of infinitely many \v Soltés' digraphs, the digraph analogue of \v Soltés' graphs. We also give an example of a \v Soltés' digraph with trivial automorphism group.
Paper Structure (5 sections, 5 theorems, 4 equations, 1 figure)

This paper contains 5 sections, 5 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proofs on the abundancy of $z$-Šoltés' digraphs
  3. Few negative Šoltes' digraphs
  4. Detailed computations
  5. A Šoltes' digraph with trivial automorphism group.

Key Result

Theorem 1

For every $z \in \mathbb Z,$ there are infinitely many digraphs $D$ for which $W(D)-W(D \setminus v)=z$ for every $v \in V.$

Figures (1)

  • Figure 1: The Šoltes' digraph $D(85,\{4\})$

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Example 3
  • Definition 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • proof : Proof of \ref{['thr:main']}
  • Claim 7
  • ...and 5 more