### Anti-Adjacency Matrices of Certain Graphs Derived from Some Graph Operations

*Manju V N, Athul T B, Suresh Singh G*

#### Abstract

If we go through the literature, one can find many matrices that are derived for a given simple graph. The one among them is the anti-adjacency matrix which is given as follows; The anti-adjacency matrix of a simple undirected graph $G$ with vertex set $V (G) \,= \,\{\,v_1,\,v_2,\\ \dots, v_n\}$ is an $n \times n$ matrix $B(G) = (b_{ij} )$, where $b_{ij} = 0$ if there exists an edge between $v_i$ and $v_j$ and $1$ otherwise. In this paper, we try to bring out an expression, which establishes a connection between the anti-adjacency matrices of the two graphs $G_1$ and $G_2$ and the anti-adjacency matrix of their tensor product, $G_1 \otimes G_2$. In addition, an expression for the anti-adjacency matrix of the disjunction of two graphs, $G_1\lor G_2$, is obtained in a similar way. Finally, we obtain an expression for the anti-adjacency matrix for the generalized tensor product and generalized disjunction of two graphs. Adjacency and anti-adjacency matrices are square matrices that are used to represent a finite graph in graph theory and computer science. The matrix elements show whether a pair of vertices in the graph are adjacent or not.

#### Keywords

Anti-adjacency matrix, tensor product, disjunction, generalized tensor product and generalized disjunction.

#### References

M. Edwina, K.A. Sugeng, Determinant of Anti-adjacency Matrix of Union and Join Operation from Two disjoint of Several Classes of Graphs, AIP Conference Proceedings 1862, 030158 (2017); https://doi.org/10.1063/1.4991262.

Frank Harary and Gordon W. Wilcox, Boolean Operations on Graphs, MATH.SCAND, 20(1967), 41-51.

G. Suresh Singh, Graph Theory, PHI, New Delhi, 2010.

V.N. Manju and G. Suresh Singh, Adjacency Matrices of Generalized Composition and Generalized Disjunction of Graphs, Advances in Mathematics: Scientific Journal 9(2020), no.3, 101-1051, ISSN: 1857-8365(Printed); 1857-838(electronic); https://doi.org/10.37418/amsj.9.3.29.

U.P. Acharya, H.S. Mehta, 2-Tensor Product of Graphs, International Journal of Mathematics and Scientific Computing (ISSN: 2231-5330), Vol. 4, No. 1, 2014.

DOI:

http://dx.doi.org/10.23755/rm.v48i0.1189
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