The Base Of The Primitive Nonpowerful Symmetric Sign Pattern Matrix

Combinatorial matrix theory, as a new branch of Mathematics, have developed rapidly on the last two decades, along with the rich study of linear algebra, graph theory and combinatorics. And Sign Pattern matrix theory is active and important components of combinatorial matrix theory. There are in plenty of applications in many fields.This thesis mainly discusses the base set of the primitive nonpowerful symmetric sign pattern matrix. Because symmetric sign pattern matrix and signed graph are closely interconnected, we will study the primitive nonpowerful symmetric sign pattern matrix by using the graph theory in this work, In this case study, we will study the Primitive nonpowerful signed barbell and the primitive nonpowerful signed simple graph, we will calculate the boundary of the base, determine the extremal graph, and characterize the base set by Structuring graphs. The major contents as following:In section 1, we introduce the research background of graph theory, combinator-ics and sign pattern matrix, explain some basic relational concepts, and state the current research situations of the signed pattern matrix at home and abroad. In section 2, we define the Primitive nonpowerful signed barbell, besides, give the important conclusions about the base, and characterize the extremal graph and base set. In sec-tion 3, we define the primitive nonpowerful signed simple graph. According to the range of base obtained, we characterize the extremal graph and base set. The last section, we summarize the research and point out the aspects that will be further studied.