Attainable bounds for algebraic connectivity and maximally-connected regular graphs

We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for very few possible girths. For diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters $D$ up to and including $D=9$ (the case of $D=10$ is open). These graphs are extremely rare and also have high girth; for example we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when $D=7$; all have girth 8 (out of a total of about $10^{20}$ cubic graphs on 44 vertices, including 266362 having girth 8). We also exhibit families of $d$-regular graphs attaining upper bounds with $D=3$ and $4$, and with $g=6.$ Several conjectures are proposed.

[1]  Jan Goedgebeur,et al.  House of Graphs 2.0: a database of interesting graphs and more , 2022, Discret. Appl. Math..

[2]  Theodore Kolokolnikov,et al.  It is better to be semi-regular when you have a low degree , 2021, ArXiv.

[3]  Madhu N. Belur,et al.  Algebraic Connectivity: Local and Global Maximizer Graphs , 2021, IEEE Transactions on Network Science and Engineering.

[4]  Sebastian M. Cioabùa,et al.  Eigenvalues of graphs and spectral Moore theorems , 2020, 2004.09221.

[5]  Norikazu Takahashi,et al.  Maximizing Algebraic Connectivity in the Space of Graphs With a Fixed Number of Vertices and Edges , 2017, IEEE Transactions on Control of Network Systems.

[6]  Hiroshi Nozaki,et al.  Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue , 2015, SIAM J. Discret. Math..

[7]  Theodore Kolokolnikov,et al.  Maximizing algebraic connectivity for certain families of graphs , 2014, ArXiv.

[8]  Carlos M. da Fonseca,et al.  The γ-Spectrum of a Graph , 2011, Ars Comb..

[9]  Alexander Lubotzky,et al.  Expander graphs in pure and applied mathematics , 2011, 1105.2389.

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

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

[12]  Brendan D. McKay,et al.  Computational determination of (3, 11) and (4, 7) cages , 2010, J. Discrete Algorithms.

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

[14]  Geoffrey Exoo,et al.  Crossing Number Graphs , 2009 .

[15]  Yoonsoo Kim,et al.  Bisection algorithm of increasing algebraic connectivity by adding an edge , 2009, 2009 17th Mediterranean Conference on Control and Automation.

[16]  Piet Van Mieghem,et al.  Algebraic connectivity optimization via link addition , 2008, BIONETICS.

[17]  Damon Mosk-Aoyama,et al.  Maximum algebraic connectivity augmentation is NP-hard , 2008, Operations Research Letters.

[18]  N. Abreu.  Old and new results on algebraic connectivity of graphs , 2007 .

[19]  Stephen P. Boyd,et al.  Finite state abstraction of a stochastic model of the lactose regulation system of Escherichia coli , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[20]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[21]  R. Olfati-Saber.  Ultrafast consensus in small-world networks , 2005, Proceedings of the 2005, American Control Conference, 2005..

[22]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[23]  A. Nilli,et al.  Tight Estimates for Eigenvalues of Regular Graphs , 2004 .

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

[25]  J. Friedman.  Some geometric aspects of graphs and their eigenfunctions , 1993 .

[26]  Joel Friedman,et al.  On the second eigenvalue and random walks in randomd-regular graphs , 1991, Comb..

[27]  N. Alon.  Eigenvalues and expanders , 1986, Comb..

[28]  A. J. Hoffman,et al.  On the line graph of a projective plane , 1965 .

[29]  H. J. Ryser,et al.  Geometries and Incidence Matrices , 1955 .

[30]  Gang Li,et al.  Maximizing Algebraic Connectivity via Minimum Degree and Maximum Distance , 2018, IEEE Access.

[31]  M. Abreu,et al.  On (minimal) regular graphs of girth 6 , 2006, Australas. J Comb..

[32]  Brendan D. McKay,et al.  Fast backtracking principles applied to find new cages , 1998, SODA '98.

[33]  D. Cvetkovic,et al.  There are exactly 13 connected, cubic, integral graphs , 1975 .

[34]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[35]  A. Hoffman.  On the Polynomial of a Graph , 1963 .