110-vertex Iofinova–Ivanov graph

Semi-symmetric cubic graph with 110 vertices and 165 edges 110-vertex Iofinova–Ivanov graph 110 vertex Iofinova Ivanov graph.svg 110 165 1320 (PGL2(11)) 7 7 10 2 3 semi-symmetricbipartitecubicHamiltonian The 110-vertex Iofinova–Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges. The graph is named after Marina Evgenievna Iofinova and Alexander A. Ivanov, who constructed this graph in 1985 alongside four other graphs, with 126, 182, 506, and 990 vertices, with special symmetries. Properties Marina Evgenievna Iofinova and Alexander A. Ivanov proved in 1985 the existence of five and only five semi-symmetric cubic bipartite graphs whose automorphism groups act primitively on each partition.Han and Lu Affine primitive groups and Semisymmetric graphs http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v20i2p39/pdf combinatorics.org 12 August 2015 The smallest has 110 vertices. The others have 126, 182, 506 and 990.Weisstein Eric Iofinova-Ivanov Graphs http://mathworld.wolfram.com/Iofinova-IvanovGraphs.html Wolfram MathWorld Wolfram 11 August 2015 The 126-vertex Iofinova–Ivanov graph is also known as the Tutte 12-cage. According to Ivanov, during their initial work on the graph, Iofinova believed that "This work will not make us famous." By 2025, their paper would go on to be Ivanov's second most cited work, followed by their paper with Cheryl E. Praeger on Affine 2-transitive graphs, although Iofinova had died in 1999.Ivanov Alexander A. Ever-Evolving Groups: An Introduction to Modern Finite Group Theory 2025 Springer Nature Switzerland Cham 978-3-031-89011-6 46 1st 2025 Symmetries The Iofinova-Ivanov graph is significant as it was specifically constructed to be acted on by a symmetry group satisfying four properties. First, the graph can partitioned into two orbits and where the stabilizer of an element in one partition has an orbit length of 3 on the elements in the other partition, and vice versa. Second, no non-trival equivalence relation on the partitions are preserved by the group. Third, any permutation on the graph that preserves the aforementioned length 3 relation will preserve the partions. Finally, the group must be self-normalizing on the permutation group on ; any permutation that normalizes the group is necessarily in the group. A group that acts on a set satisfying the mentioned properties must be one of five groups, up to isomorphism: , , , , and . Here, denotes the Projective general linear group with entries in the finite field of order , is the Projective special linear group again with entries in the corresponding field, while is the Dickson group, and is the automorphism group of the Mathieu group . The corresponding sets that these groups act on are graphs with 110, 126, 182, 506, and 990 vertices respectively, and the Iofinova-Ivanov graph corresponds to the smallest of these graphs. Algebraic properties 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.Iofinova and Ivanov Investigations in Algebraic Theory of Combinatorial Objects 2013 Springer 470 9789401719728 https://books.google.com/books?id=NVDqCAAAQBAJ&q=Iofinova+graph&pg=PA470 12 August 2015 Semi-symmetry 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. Conder M. Marston Conder Malnič A. Marušič D. Dragan Marušič Pisanski T. Tomaž Pisanski Potočnik P. 845 Ljubljana IMFM Preprints Institute of Mathematics, Physics and Mechanics The Ljubljana Graph http://www.imfm.si/preprinti/PDF/00845.pdf 40 2002 Conder Marston Marston Conder Malnič Aleksander Marušič Dragan Dragan Marušič Potočnik Primož A census of semisymmetric cubic graphs on up to 768 vertices Journal of Algebraic Combinatorics 23 2006 255–294 10.1007/s10801-006-7397-3 3 free. It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices. References Bibliography Iofinova, M. E. and Ivanov, A. A. Bi-Primitive Cubic Graphs. In Investigations in the Algebraic Theory of Combinatorial Objects. pp. 123–134, 2002. (Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, pp. 137–152, 1985.) Ivanov, A. A. Computation of Lengths of Orbits of a Subgroup in a Transitive Permutation Group. In Methods for Complex System Studies. Moscow: VNIISI, pp. 3–7, 1983. Ivanov, A. V. On Edge But Not Vertex Transitive Regular Graphs. In Combinatorial Design Theory (Ed. C. J. Colbourn and R. Mathon). Amsterdam, Netherlands: North-Holland, pp. 273–285, 1987. Category:Individual graphs Category:Regular graphs