Lyca DC. Marcelino, Isagani B. Jos, On the nullity of some families of r-partite graphs, Eur. J. Math. Appl. 2 (2022), Article ID 16.

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Volume 2 (2022), Article ID 16

https://doi.org/10.28919/ejma.2022.2.16

Published: 06/12/2022

Abstract:

The nullity of a graph $G$, denoted by $\eta(G)$ is defined to be the multiplicity of the eigenvalue zero in the spectrum of a graph. The spectrum of a graph $G$ is a two-row matrix, the first row elements are the distinct eigenvalues of its adjacency matrix $A(G)$ and the second row elements are its corresponding multiplicities. Furthermore, the rank of $G$, denoted by $rank(G)$ is also the rank of $A(G)$, that is $rank(G) = rank(A(G))$. In addition, given that $G$ is of order $n$, it is known that $\eta(G) = n – rank(G)$. Thus, any result about rank can be stated in terms of nullity and vice versa. In this paper, we investigate some families of $r$-partite graphs of order $n$ and we determine the nullity of these $r$-partite families using its rank. First, we consider the complete $r$-partite graphs denoted by $K_{n_1,n_2,n_3,…,n_r}$ where $n = n_1+n_2+n_3+…+n_r$ and $r \geq 4$. Second, we also consider a family of $r$-partite graphs where $n \geq 2r – 1$ and $r \geq 4$, which is an extension of a family of tripartite graphs introduced in the paper “On the nullity of a family of tripartite graphs” by Farooq, Malik, Pirzada and Naureen.

How to Cite:

Lyca DC. Marcelino, Isagani B. Jos, On the nullity of some families of r-partite graphs, Eur. J. Math. Appl. 2 (2022), Article ID 16. https://doi.org/10.28919/ejma.2022.2.16