The Relative Extrema of Vertex Degree Distances of Dumbbell Graphs

Abstract

The dumbbell graph is defined as a graph joining two cycles with the end-vertices of a path. In this paper, the vertex degree distances of the dumbbell graph, and the distribution of the relative extrema of vertex degree distances on the path and on two cycles of the dumbbell graph are studied. Some meaningful results are obtained: (1) on the path, the vertex degree distance is maximized at the end-vertex joining with the cycle of less length and minimized at the vertex nearing the end-vertex joining with the cycle of longer length; (2) on the cycles except end-vertices (of the path), the minimum of the vertex degree distance is taken at the neighbor vertices of end-vertex of the less length cycle; the maximum is taken at the vertex farthest from the end-vertex, but whether the vertex of the maximum is on the long cycle or the short cycle is related not only to the two cycle length, but also to the path length.