Complete characterization of bicyclic minimal configurations
Xuezhong Tan, Bo Liu · Linear and Multilinear Algebra · 2012
Sciriha first proposed the notion of minimal configurations. A graph G with nullity one is called a minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. Recently, minimal configurations in special class of graphs, such as trees and unicyclic graphs, have been studied. The set of bicyclic graphs, denoted by , can be partitioned into two subsets: the set ℬ* of graphs which contain induced ∞-graphs, and the set ℬ** of graphs which contain induced θ-graphs. Nath described the minimal configurations in ℬ*. In this article, we will describe the minimal configurations in the set ℬ**. These results together give a complete characterization of the minimal configurations in .