Several properties of antiadjacency matrices of directed graphs

Abstract

Let $ G $ be a directed graph with $ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\; n. $ The adjacency matrix of the directed graph $ G $ is a matrix $ A = \left[{a}_{ij}\right] $ of order $ n\times n, $ such that for $ i\ne j $, if there is an arc from $ i $ to $ j, $ then $ {a}_{ij} = 1 $, otherwise $ {a}_{ij} = 0 $. Matrix $ B = J-A $ is called the antiadjacency matrix of the directed graph $ G, $ where $ J $ is the matrix of order $ n\times n $ with all of those entries are one. In this paper, we provided several properties of the adjacency matrices of directed graphs, such as a determinant of a directed graphs, the characteristic polynomial of acyclic directed graphs, and regular directed graphs. Moreover, we discuss antiadjacency energy of acyclic directed graphs and give some examples of antiadjacency energy for several families of graphs.

Keywords

• antiadjacency matrix
• characteristic polynomial
• directed acyclic graph
• energy of directed graph
• spectrum