On Graphs with the Smallest Eigenvalue at Least $-1-\sqrt{2}$, part II

TL;DR

This paper continues the study of graphs with a lower bound on their smallest eigenvalue, identifying all minimal graphs outside a specific family as isomorphic to one of 38 computer-found graphs.

Contribution

It extends previous work by classifying minimal graphs not in the H-family, showing they are among 38 specific graphs identified computationally.

Findings

Identified 38 minimal graphs outside the H-family.

Confirmed these graphs are the only minimal exceptions.

Extended the classification of graphs with eigenvalue bounds.

Abstract

This is a continuation of the article with the same title. In this paper, the family H is the same as in the previous paper "On Graphs with the Smallest Eigenvalue at Least $-1-\sqrt{2}$, part I". The main result is that a minimal graph which is not an H -line graph, is just isomorphic to one of the 38 graphs found by computer.