On \(P_{3}\)-Degree of Graphs

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 \( n \geq 6 \) there is at least one graph of order \( n \), with distinct \( P_3 \)-degree vertices.