  The neighbourhood matrix, $\mathcal{NM}(G)$, a novel representation of graphs
proposed in \cite {ALPaper} is defined using the neighbourhood sets of the
vertices. The matrix also exhibits a bijection between the product of two
well-known graph matrices, namely the adjacency matrix and the Laplacian
matrix. In this article, we extend this work and introduce the sequence of
powers of $\mathcal{NM}(G)$ and denote it by $ \mathcal{NM}^{\{l\}}, 1\leq l
\leq k(G) $ where $ k(G) $ is called the \textbf{iteration number}, $
k(G)=\lceil{\log_{2} diameter(G)}\rceil$. The sequence of matrices captures the
distance between the vertices in a profound fashion and is found to be useful
in various applications. One of the interesting results of this article is that
whenever $ \eta_{ij}^{\{l\}}=-1$, for $ 1\leq l \leq k(G) $, then
$d_{G}(i,j)=2^{l}$ , where $d_{G}(i,j)$ is the shortest path distance between $
i $ and $ j $. Further, we characterize the entries of the matrices $
\mathcal{NM}^{\{l\}}$, for every $l, 1\leq l \leq k(G)$. Using this concept of
the sequence of powers of neighbourhood matrix and with the aid of some of its
properties, we propose an algorithm to find the shortest path between any pair
of vertices in a given undirected unweighted simple graph. The proposed
algorithm and the claims therein are formally validated through simulations on
synthetic data and the real network data from Facebook where sampling-based
computations are performed for large collection of graphs containing
large-sized graph. The empirical results are quite promising with our algorithm
having the best running time among all the existing well known shortest path
algorithms.
