On Edge-graceful Spectra of Square of Paths with Odd Order

For a simple path Pr on r vertices, the square of Pr is the graph P 2 r on the same set of vertices of Pr, and where every pair of vertices of distance 2 or less in Pr are connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and an integer k ≥ 0, G is said to be k-edge-graceful if the edges can be labeled by k, k+1, . . . , k+q−1 so that the induced vertex sums (mod p) are distinct, where the vertex sum at a vertex is the sum of the labels of all edges incident to such vertex. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI(G). In this article, the edge-graceful spectrum egI(P 2 r ) for square of paths is completely determined for odd r.