A new achievement from reviewing some of the content about the edge-distance-balanced property of the generalized Petersen graphs GP(4i + 7, 2)

Abstract

In this paper, we initially introduce the concept of edge-distance-balanced property which is considered as the generalized concept of distance-balanced property. In our consideration, we will discover some interesting properties of generalized Petersen graphs that called edge distance-balanced graphs. We also define a connection between edge-distance-balanced graphs and distance-balanced graphs. Furthermore, a graph G is said to be edge-distance-balanced if for any edge uv of G, the number of edges closer to u than to v is equal to the number of edges closer to v than to u. It will be concluded that for any round number i ≥ 3, the generalized Petersen graph GP(4i + 7, 2) is not edge-distance-balanced.

Subject Classification:

• 05C38
• 05C12

Keywords:

• Edge-distance-balanced graph
• Generalized Petersen graph
• Distance-balanced graph