A note on edge-graceful spectra of the square of paths

For a simple path PrPr on rr vertices, the square of PrPr is the graph Pr2 on the same set of vertices of PrPr, and where every pair of vertices of distance two or less in PrPr is connected by an edge. Given a (p,q)(p,q)-graph GG with pp vertices and qq edges, and a nonnegative integer kk, GG is said to be kk-edge-graceful if the edges can be labeled bijectively by k,k+1,…,k+q−1k,k+1,…,k+q−1, so that the induced vertex sums (modp) are pairwise distinct, where the vertex sum (modp) at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices pp. We call the set of all such kk the edge-graceful spectrum of GG, and denote it by egI(G)egI(G). In this article, the edge-graceful spectrum egI(Pr2) for the square of paths Pr2 is completely determined for odd rr.