Determining the Number of Connected Vertices Labelled Graph of Order Five with Maximum Number of Parallel Edges is Five and Containing No Loops

A connected graph is a graph where there exists at least a path joining every pair of the vertices in the graph, and a graph is called simple if that graph containing no loops nor parallel edges. Given a graph G (V, E) with n vertices and m edges, there are a lot of graph that can be formed, either connected or disconnected, or simple or not simple. In this paper we will discuss the number of connected vertex labelled graph of order five (n=5) and 4 ≤ m ≤ 10, with maximum number of parallel edges that connecting different pairs of vertices is five (the parallel edges that connecting the same pair of vertices are counted as one).