Final answer:
The girth and diameter differ among the graphs: Kn (girth undefined for n<3, otherwise 3; diameter 1), Km,n (girth 4 if m,n>=2; diameter 2), Petersen (girth 5; diameter 2), Dodecahedron (girth 5; diameter 5), and 5D-cube (girth 4; diameter 5).
Step-by-step explanation:
The question asks to determine the girth and the diameter of several types of graphs: Kn (complete graph on n vertices), Km,n (complete bipartite graph on m+n vertices), Petersen graph, Dodecahedron graph, and the 5-dimensional cube (5D-cube).
For Kn, the girth is not defined as there are no cycles in a complete graph if n < 3. For n >= 3, the girth (the length of the shortest cycle) is 3 since any three vertices will form a triangle. The diameter (the greatest distance between any pair of vertices) is 1, as every vertex is connected to every other vertex directly.
For Km,n, the girth is also 4 if both m and n are >= 2, as there will be a four-vertex cycle involving two vertices from each partition. The diameter is always 2, as any vertex is connected to any other vertex through one intermediary at most.
The Petersen graph has a girth of 5, which is the length of its shortest cycle, and the diameter is 2, the greatest distance between any two vertices.
The Dodecahedron graph has a girth of 5, representing the shortest cycle corresponding to its pentagonal faces, and a diameter of 5, which is the maximal number of edges in the shortest path between any two vertices.
Finally, the 5D-cube, also known as the 5-dimensional hypercube, has a girth of 4 as with any hypercube, since the smallest cycles are formed by the squares at the edges of the hypercube. The diameter of the 5D-cube is 5, the number of dimensions, as this is the number of steps needed to get from one vertex to the opposite vertex.