# Determinant of antiadjacency matrix of union and join operation from two disjoint of several classes of graphs

M. Edwina, K. A. Sugeng

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Citation (Scopus)

## Abstract

Let G be a graph with V(G) = {v1, ..., vn} and E(G) = {e1, ...,em}. We only consider undirected graphs with no multiple edges in this paper. The adjacency matrix of G, denoted by A(G), is the n × n matrix A = [aij], where aij = 1 if e = vivj ∈ E(G) or otherwise aij = 0. The anti adjacency matrix of G, denoted by B(G), is the n × n matrix B = [bij], where bij = 0 if e = vivj ∈ E(G) or otherwise bij = 1. Properties of the determinant of the adjacency matrix of some simple graphs have been studied by many researchers. However, the determinant of the anti-adjacency matrix has not been explored yet. If G1 and G2 are disjoint graphs, then the joining of two graphs G1 and G2, denoted G1 G2 is defined by taking copies of G1 and G2 and adding edges so that each vertex in G1 is adjacent to every vertex in G2. In this paper, we show the properties of the determinant of joining two graphs, G1 and G2. Union of two graphs, denote G1 ∪ G2 is a graph formed by taking copies of G1 and G2. The objectives of this paper are to identify some properties of the determinant anti adjacency matrix of joining and union operation from two disjoint graphs. This paper also emphasizes on investigating the determinant of some special graph class formed by joining and unioning operation of two disjoint of several classes of graphs, such as Bipartite graphs, Cycles, Complete graphs, Stars, and Wheels.

Original language English International Symposium on Current Progress in Mathematics and Sciences 2016, ISCPMS 2016 Proceedings of the 2nd International Symposium on Current Progress in Mathematics and Sciences 2016 Kiki Ariyanti Sugeng, Djoko Triyono, Terry Mart American Institute of Physics Inc. 9780735415362 https://doi.org/10.1063/1.4991262 Published - 10 Jul 2017 2nd International Symposium on Current Progress in Mathematics and Sciences 2016, ISCPMS 2016 - Depok, Jawa Barat, IndonesiaDuration: 1 Nov 2016 → 2 Nov 2016

### Publication series

Name AIP Conference Proceedings 1862 0094-243X 1551-7616

### Conference

Conference 2nd International Symposium on Current Progress in Mathematics and Sciences 2016, ISCPMS 2016 Indonesia Depok, Jawa Barat 1/11/16 → 2/11/16

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