# Generating new graphs using Boolean operations (V and λ) on adjacency and antiadjacency matrices of graphs

Gisca A.T.A. Putri, Wismoyo Adinegoro, Kiki Ariyanti

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

## Abstract

Let G be a graph with V(G) = {v1, ., vn} and E(G) = {e1, ., em}. We only consider 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 = vi vj ϵ E(G) or aij = 0 otherwise. The antiadjacency matrix of G, denoted by B(G), is the n × n matrix B = [bij], where bij = 0 if e = vi vj ϵ E(G) or bij = 1 otherwise. Harary and Wilcox have considered Boolean operations on two graphs G1 and G2, resulting a new graph G whose V(G) equals V(G1) × V(G2). On this paper, Boolean operations are defined for two adjacency and two antiadjacency matrices of graphs Gl and G2 with V(G1) = V(G2) rather than looking at the two graphs themselves. Boolean operations which are reviewed on this paper are V and λ. The objectives of this paper are to introduce the operations on two adjacency or two antiadjacency matrices of graph G, to discover some characteristics of the operations on the matrices, to construct a new graph which is generated using the operators on two adjacency or antiadjacency matrices, and to reveal the similarity between operators on adjacency matrix and operators on antiadjacency matrix based on the represented graph. This paper also emphasizes on investigating the relationship between the operators and on comparing the largest eigenvalue between graphs which are constructed by Boolean operators on both adjacency and antiadjacency matrices.

Original language English International Symposium on Current Progress in Mathematics and Sciences 2015, ISCPMS 2015 Proceedings of the 1st International Symposium on Current Progress in Mathematics and Sciences Terry Mart, Djoko Triyono American Institute of Physics Inc. 9780735413764 https://doi.org/10.1063/1.4946905 Published - 19 Apr 2016 1st International Symposium on Current Progress in Mathematics and Sciences, ISCPMS 2015 - Depok, IndonesiaDuration: 3 Nov 2015 → 4 Nov 2015

### Publication series

Name AIP Conference Proceedings 1729 0094-243X 1551-7616

### Conference

Conference 1st International Symposium on Current Progress in Mathematics and Sciences, ISCPMS 2015 Indonesia Depok 3/11/15 → 4/11/15