Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.

[1]  Vito Napolitano,et al.  A family of regular graphs of girth 5 , 2008, Discret. Math..

[2]  Brendan D. McKay,et al.  The Smallest Cubic Graphs of Girth Nine , 1995, Combinatorics, Probability and Computing.

[3]  Camino Balbuena,et al.  On the connectivity of cages with girth five, six and eight , 2007, Discret. Math..

[4]  Geoffrey Exoo.  A Simple Method for Constructing Small Cubic Graphs of Girths 14, 15, and 16 , 1996, Electron. J. Comb..

[5]  Vito Napolitano,et al.  Íëìêêääëááae Âçíêaeaeä Ç Çååáaeaeìçêááë Îóðùññ ¿½´¾¼¼¼µ¸è× ½½½ß¾¼¼ , 2022 .

[6]  Camino Balbuena,et al.  Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages , 2010, Discret. Math..

[7]  Stanley E. Payne,et al.  Affine representations of generalized quadrangles , 1970 .

[8]  W. T. Tutte.  A family of cubical graphs , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[10]  Felix Lazebnik,et al.  Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size , 1995, Discret. Appl. Math..

[11]  M. O'Keefe,et al.  The smallest graph of girth 6 and valency 7 , 1981, J. Graph Theory.

[12]  Camino Balbuena,et al.  A Construction of Small (q-1)-Regular Graphs of Girth 8 , 2015, Electron. J. Comb..

[13]  Camino Balbuena,et al.  A construction of small regular bipartite graphs of girth 8 , 2009 .

[14]  Camino Balbuena,et al.  Families of small regular graphs of girth 5 , 2011, Discret. Math..

[15]  Felix Lazebnik,et al.  New upper bounds on the order of cages , 1996, Electron. J. Comb..

[16]  Camino Balbuena,et al.  Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices , 2008, SIAM J. Discret. Math..

[17]  Markus Meringer,et al.  Fast generation of regular graphs and construction of cages , 1999, J. Graph Theory.

[18]  Walter Feit,et al.  The nonexistence of certain generalized polygons , 1964 .

[19]  Dragan Marusic,et al.  The 10-cages and derived configurations , 2004, Discret. Math..

[20]  V. A. Ustimenko,et al.  A linear interpretation of the flag geometries of Chevalley groups , 1990 .

[21]  G. Exoo,et al.  Dynamic Cage Survey , 2011 .

[22]  Derek Allan Holton,et al.  The Petersen graph , 1993, Australian mathematical society lecture series.

[23]  András Gács,et al.  On geometric constructions of (k, g)-graphs , 2008, Contributions Discret. Math..

[24]  Pak-Ken Wong,et al.  Cages - a survey , 1982, J. Graph Theory.

[25]  Camino Balbuena,et al.  Constructions of small regular bipartite graphs of girth 6 , 2011, Networks.

[26]  E. Bannai,et al.  On finite Moore graphs , 1973 .

[27]  H. Sachs,et al.  Regukre Graphen gegebener Taillenweite mit minimaler Knotenzahl , 1963 .

[28]  Norman Biggs,et al.  Constructions for Cubic Graphs with Large Girth , 1998, Electron. J. Comb..

[29]  C. Balbuena,et al.  A formulation of a (q+1,8)-cage , 2015, 1501.02448.

[30]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[31]  Jacques Tits,et al.  Sur la trialité et certains groupes qui s’en déduisent , 1959 .

[32]  J. Thas,et al.  Finite Generalized Quadrangles , 2009 .

[33]  C. T. Benson.  Minimal Regular Graphs of Girths Eight and Twelve , 1966, Canadian Journal of Mathematics.

[34]  R. M. Damerell.  On Moore graphs , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.