Isoperimetric Constants of Infinite Plane Graphs

AbstractLet G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be infHl(H) / a(H) and define the isoperimetric constant h (δ) to be infG h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 : $$ {{(\delta -6)(\delta^2 -8 \delta + 15)}\over {(\delta-4)(\delta^2 - 8\delta + 13)}} \le h (\delta) \le \sqrt {{\delta-6}\over {\delta-2}}. $$

[1]  Russell Lyons,et al.  Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model , 2000, math/0008191.

[2]  B. Mohar.  Isoperimetric inequalities, growth, and the spectrum of graphs , 1988 .

[3]  Branko Grünbaum,et al.  Edge-transitive planar graphs , 1987, J. Graph Theory.

[4]  Michael D. Plummer,et al.  Genus bounds for embeddings with large minimum degree and representativity , 2002, Discret. Math..

[5]  Shing-Tung Yau,et al.  Isoperimetric constants and the first eigenvalue of a compact riemannian manifold , 1975 .

[6]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[7]  Andrzej Żuk,et al.  On the norms of the random walks on planar graphs , 1997 .

[8]  J. Shawe-Taylor,et al.  The Spectral Radius of Infinite Graphs , 1988 .

[9]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[10]  David R. DeBaun,et al.  cohomology of noncompact surfaces , 1984 .

[11]  B. Mohar,et al.  A Survey on Spectra of infinite Graphs , 1989 .

[12]  Paolo M. Soardi,et al.  Recurrence and transience of the edge graph of a tiling of the euclidean plane , 1990 .

[13]  Bojan Mohar,et al.  Embeddings of infinite graphs , 1987, J. Comb. Theory B.

[14]  J. Ratcliffe.  Foundations of Hyperbolic Manifolds , 2019, Graduate Texts in Mathematics.

[15]  Wolfgang Woess,et al.  A note on tilings and strong isoperimetric inequality , 1998, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  T. Shirai,et al.  Isoperimetric Constants of (d,f) Regular Planar Graphs , 2003 .

[17]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .