TITLE:
Graphs with Pendant Vertices and r(G) ≤ 7
AUTHORS:
Haicheng Ma, Shang Gao, Danyang Li
KEYWORDS:
Adjacency Matrix, Rank, Nullity, Multiplication of Vertices
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.8 No.2,
January
21,
2020
ABSTRACT: Let G be a graph of order n with vertex set V(G) = {v1, v2,…, vn}. the adjacency matrix of G is an n × n matrix A(G) = (aij)n×n, where aij is the number edges joining vi and vj in G. The eigenvalues λ1, λ2, λ3,…, λn, of A(G) are said to be the eigenvalues of the graph G and to form the spectrum of this graph. The number of nonzero eigenvalues and zero eigenvalues in the spectrum of G are called rank and nullity of the graph G, and are denoted by r(G) and η(G), respectively. It follows from the definitions that r(G) + η(G) = n. In this paper, by using the operation of multiplication of vertices, a characterization for graph G with pendant vertices and r(G) = 7 is shown, and then a characterization for graph G with pendant vertices and r(G) less than or equal to 7 is shown.