Conjugate Laplacian eigenvalues of co-neighbour graphs

Abstract

Let \(G\) be a simple graph of order \(n\). A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in \(G\) are co-neighbour vertices if they share the same neighbours. Clearly, if \(S\) is a set of pairwise co-neighbour vertices of a graph \(G\), then \(S\) is an independent set of \(G\). Let \(c = a + b\sqrt{m}\) and \(\overline{c} = a-b\sqrt{m}\), where \(a\) and \(b\) are two nonzero integers and \(m\) is a positive integer such that \(m\) is not a perfect square. In [M. Lepovi\'c, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730--738,  2007], the author defined the matrix \(A^c(G) = [c_{ij} ]_n\) to be the conjugate adjacency matrix of \(G,\) if \(c_{ij}=c\) for any two adjacent vertices \(i\) and \(j\), \(c_{ij}= \overline{c}\) for any two nonadjacent vertices \(i\) and \(j\), and \(c_{ij}= 0\) if \(i=j\). In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices.