Fivevertex

Fivevertex is a term used in graph theory to refer to graphs with five vertices. It may denote the general class of five-vertex graphs or serve as a label for datasets or software focused on these graphs. The concept is used to study basic questions about connectivity, degree sequences, and isomorphism in a compact setting.

On five vertices there are 2^10 = 1024 labeled simple graphs, and 34 nonisomorphic graphs up to isomorphism. Notable examples include K5, the complete graph on five vertices; C5, the cycle graph of length five; P5, the path on five vertices; K1,4, the star with a central vertex connected to four leaves; and K2,3, the complete bipartite graph with partitions of sizes 2 and 3. K5 is nonplanar by Kuratowski’s theorem, while C5 is planar; many other five-vertex graphs are also planar. Degree sequences on five vertices range from (0,0,0,0,0) to (4,1,1,1,1) and others, reflecting the graph’s edge count from 0 up to 10.

Applications of fivevertex-scale study include teaching basic graph theory, benchmarking algorithms for isomorphism and planarity, and serving as a compact testbed for graph-enumeration techniques. Some online resources organize the set of five-vertex graphs as catalogs or databases to facilitate quick reference and research.