New strongly regular graphs derived from the G2(4) graph

TL;DR

This paper constructs and analyzes new strongly regular graphs derived from the G2(4) graph, revealing novel properties and subgraphs, including five new strongly regular graphs with specific parameters.

Contribution

The paper explicitly constructs the G2(4) graph and a subgraph E, then derives five new strongly regular graphs, some of which are completely new discoveries.

Findings

Discovered five strongly regular graphs from the subgraph E

Identified a subgraph F isomorphic to known structures, with overlooked strong regularity

Provided computational verification using specialized graph software

Abstract

We consider simple loopless finite undirected graphs. Such a graph is called strongly regular with parameter set (v,k,l,m), for short a srg(v,k,l,m), iff it has exactly v vertices, each of them has exactly k neighbours, and the number of common neighbours of any two different vertices is l if they are neighbours and m otherwise. The G2(4) graph is a well-known srg(416,100,36,20). In this article, we explicitly construct it and a certain subgraph E induced by 320 vertices in the same way as in an older article by this author. We discover some interesting properties of E and derive five strongly regular graphs from it: A srg(256,60,20,12) F which is a subgraph induced by 256 vertices and four srg(336,80,28,16) H, H_1, H_2 and H_3 which do have E as induced subgraph. The latter three graphs are new in version 4 of this article and seem to have been completely unknown as H was before version 1 appeared. The graph F is isomorphic to objects described as unions of 16 16-cocliques in a description of subgraphs of the G2(4) graph by Andries E. Brouwer; but the strong regularity has been unnoticed before version 1 of this article. Several propositions in this article have been checked by executing the additionally (in the source package) provided program G24DGS2 and the program Dreadnaut from the popular graph theoretic software nauty (by Brendan McKay and Adolfo Piperno).