A Graph-Theoretical Proof of an Adjacency Property of Major Submatrices

Abstract

Major submatrices of an $n \times m( {n\leqq m} )$ matrix of rank n have an important role in the theory of a linear graph. This paper presents the graph-theoretical relationships among the major submatrices. “Adjacency” of two major submatrices is defined. Based on this new concept, a linear graph called a K-graph is defined, which represents the adjacencies among the major submatrices of a given matrix. The existence of a Hamilton circuit is shown in a K-graph. The proof can provide an efficient procedure for obtaining all the major submatrices.