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

For a simple path image on image vertices, the square of image is the graph image on the same set of vertices of image, and where every pair of vertices of distance two or less in image is connected by an edge. Given a image-graph image with image vertices and image edges, and a nonnegative integer image, image is said to be image-edge-graceful if the edges can be labeled bijectively by image, so that the induced vertex sums image are pairwise distinct, where the vertex sum image at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices image. We call the set of all such image the edge-graceful spectrum of image, and denote it by image. In this article, the edge-graceful spectrum image for the square of paths image is completely determined for odd image.