On the nullity of some families of r-partite graphs

The nullity of a graph G, denoted by η(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)), which is defined as the maximum number of linearly independent row/column vectors in A(G). In addition, it is known that η(G) = n − rank(G), thus any result about rank can be stated in terms of nullity and vice versa. In this paper, we investigate three different families of r-partite graphs of order n and we determine the nullity of these r-partite families using its rank. First, a complete r-partite graphs denoted by Kn1,n2,n3,...,nr where n = n1 + n2 + n3 + ... + nr and r ≥ 4. Second, the family of r-partite graphs where n ≥ 2r − 1 and r ≥ 4 and is an extension of family of tripartite graphs introduced in the paper “On the nullity of a family of tripartite graphs” by Farooq, Malik, Pirzada and Naureen. While the third one is another family of r-partite graphs where n ≥ (2 r 2+r 2) and r ≥ 4. We characterize the third family with r-partition that satisfy properties different from what we obtain in the second family of r- partite graphs