| 110-vertex Iofinova-Ivanov graph | |
|---|---|
 | |
| Vertices | 110 |
| Edges | 165 |
| Radius | 7 |
| Diameter | 7 |
| Girth | 10 |
| Automorphisms | 1320 (PGL2(11)) |
| Chromatic number | 2 |
| Chromatic index | 3 |
| Properties | semi-symmetric bipartite cubic Hamiltonian |
| Table of graphs and parameters | |
The 110-vertex Iofinova-Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges.
Iofinova and Ivanov proved in 1985 the existence of five and only five semi-symmetric cubic bipartite graphs whose automorphism groups act primitively on each partition. [1] The smallest has 110 vertices. The others have 126, 182, 506 and 990. [2] The 126-vertex Iofinova-Ivanov graph is also known as the Tutte 12-cage.
The diameter of the 110-vertex Iofinova-Ivanov graph, the greatest distance between any pair of vertices, is 7. Its radius is likewise 7. Its girth is 10.
It is 3-connected and 3-edge-connected: to make it disconnected at least three edges, or at least three vertices, must be removed.
The chromatic number of the 110-vertex Iofina-Ivanov graph is 2: its vertices can be 2-colored so that no two vertices of the same color are joined by an edge. Its chromatic index is 3: its edges can be 3-colored so that no two edges of the same color met at a vertex.
The characteristic polynomial of the 110-vertex Iofina-Ivanov graph is . The symmetry group of the 110-vertex Iofina-Ivanov is the projective linear group PGL2(11), with 1320 elements. [3]
Few graphs show semi-symmetry: most edge-transitive graphs are also vertex-transitive. The smallest semi-symmetric graph is the Folkman graph, with 20 vertices, which is 4-regular. The three smallest cubic semi-symmetric graphs are the Gray graph, with 54 vertices, this the smallest of the Iofina-Ivanov graphs with 110, and the Ljubljana graph with 112. [4] [5] It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices.
Dragan Marušič is a Slovene mathematician. Marušič obtained his BSc in technical mathematics from the University of Ljubljana in 1976, and his PhD from the University of Reading in 1981 under the supervision of Crispin Nash-Williams.
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.
In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism
In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2.
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
In the mathematical field of graph theory, a graph G is symmetric if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism
In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of the graph's edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the first into the second.
In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive.
In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter.
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith.
In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges.
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it.
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
In the mathematical field of graph theory, the Folkman graph, named after Jon Folkman, is a bipartite 4-regular graph with 20 vertices and 40 edges.
In the mathematical field of graph theory, the Nauru graph is a symmetric bipartite cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru.
In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges.
In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges.
In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges named after W. T. Tutte.
In the mathematical field of graph theory, the Dejter graph is a 6-regular graph with 112 vertices, 336 edges and girth 6. The Dejter graph is obtained by deleting a copy of the Hamming code of length 7 from the binary 7-cube.
In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.
