The Crossing Number of Join Product of kth Power of Path Pm with Isolated Vertices and Path Pn

A graph G is said to have a crossing if two edges of G share an interior point. The minimum crossing number of G is denoted by cr(G). The crossing number problem is to find the minimum crossing solution of a graph, and it can be used in applications of circuit layout. Although the crossing numbers of join product graphs have been extensively studied, the crossing number of join product of power graphs with path is relatively unexplored. Let Pm and Pn be paths with m and n vertices, and Dn be a graph consisting of n isolated vertices. In this paper, we investigate the crossing number of kth power of path Pm that joins with isolated vertices Dn and path Pn. We have proved the minimum crossing numbers of Pkm+Dn for m ≤ 6, n ≥ 1, and Pkm+Pn for m ≤ 6, n ≥ 2.