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Graph shadows and edge-regular graphs

Jared DeLeo

TL;DR

In [1], a few methods of constructing new graphs from old are of use, and one of these is the unary"graph shadow"operation, which is generalized, and then generalized again, and conditions are given under which application of the new operations to edge-regular graphs result in edge-regular graphs.

Abstract

The definition of edge-regularity in graphs is a relaxation of the definition of strong regularity, so strongly regular graphs are edge-regular and, not surprisingly, the family of edge-regular graphs is much larger and more diverse than that of the strongly regular. In [1], a few methods of constructing new graphs from old are of use. One of these is the unary "graph shadow" operation. Here, this operation is generalized, and then generalized again, and conditions are given under which application of the new operations to edge-regular graphs result in edge-regular graphs. Also, some attention to strongly regular graphs is given.

Graph shadows and edge-regular graphs

TL;DR

In [1], a few methods of constructing new graphs from old are of use, and one of these is the unary"graph shadow"operation, which is generalized, and then generalized again, and conditions are given under which application of the new operations to edge-regular graphs result in edge-regular graphs.

Abstract

The definition of edge-regularity in graphs is a relaxation of the definition of strong regularity, so strongly regular graphs are edge-regular and, not surprisingly, the family of edge-regular graphs is much larger and more diverse than that of the strongly regular. In [1], a few methods of constructing new graphs from old are of use. One of these is the unary "graph shadow" operation. Here, this operation is generalized, and then generalized again, and conditions are given under which application of the new operations to edge-regular graphs result in edge-regular graphs. Also, some attention to strongly regular graphs is given.

Paper Structure

This paper contains 4 sections, 8 theorems, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. $(m,x)$-Shadows
  3. $(m,X)$-Shadows
  4. Generalized Graph Shadows

Key Result

Theorem 2.1

Given $G \in ER(n,d,\lambda)$, then for $x > 0$ and $m > 1$, $D_m^x(G)$ is edge- regular if and only if the following conditions hold for some nonnegative integers $d_x, \lambda_x$: 1. For all $v \in V(G)$, $|N^x(v)| = d_x$. 2. For all $u,v \in V(G)$ such that $u \sim v$ in $G$, $|N^x(u) \cap N^x(v)

Figures (6)

  • Figure 1: Distance 2 vertices from one vertex of $C_4$ to 2 other copies of $C_4$ (left). Edges are added between distance 2 vertices in different copies of $C_4$ to obtain $D_3^2(C_4)$ (right).
  • Figure 2: $C_5 \in ER(5,2,0)$ (left) and $D_2^2(C_5)$ which is not edge-regular (right).
  • Figure 3: An example of a 0-distance shadow, $D_3^0(K_3) \in ER(9,4,1)$.
  • Figure 4: $D_2^2(P_3) \cong C_6 \in ER(6,2,0)$
  • Figure 5: Distance 1 and 2 vertices from one vertex of $P_4$ to another copy of $P_4$ (left). Edges added between distance 1 and 2 vertices in different copies of $P_4$ to obtain $D_2^{1,2}(P_4)$ (right).
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Proposition 2.1
  • Corollary 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.2
  • proof
  • Definition 3.1
  • ...and 7 more