On $P_3$-Degree of Graphs

Abstract

It is known that there is not any non-trivial graph with vertices of distinct degrees, and any non-trivial graph must have at least two vertices of the same degree. In this article, we will consider the concept of $P_3$-degree of vertices and will introduce a class of connected graphs with exactly two vertices of the same $P_3$-degree. Also, the graphs with distinct $P_3$-degree vertices will be constructed and it will be proven that for any $n\ge 6$ there is at least one graph of order $n$, with distinct $P_3$-degree vertices.