Isoperimetric Constants of (d,f)-Regular Planar Graphs

Abstract

For a ðd; f Þ-regular planar graph, which is an infinite planar graph embedded in the plane such that the degree of each vertex is d and the degree of each face is f , we determine two kinds of isoperimetric constants in concrete form.

Asymptotic connectivity of hyperbolic planar graphs

“…Using the same notation as in Case 3, we have position lists π (v(2, k)) = (0, 0, 1, 1), π (v(4, k)) = (0, 1 2 , 1, 2), π (v(3, k)) = (0, 0, 1, 3 2 ), and π (v(5, k)) = (0, 1, 1, 5 2 ). The matrix A = A (4,5) governing the growth of s k is      As before lim k→∞ s k+1 s k = λ and lim k→∞ s k+1 b k = λ − 1. We partition the ball B k into three sets of vertices, B k−2 , S k−1 , and S k , satisfying s k−1 ≈ (λ − 1)b k and s k ≈ (λ − 1)λb k for large k. The list below indicates the effective degree (relative to B k ) of each type of vertex given above: degree 1: types v(2, k) and v(3, k) degree 2: types v(4, k) and v(5, k) degree 3: types v(2, k − 1), v(3, k − 1), and v(4, k − 1) degree 4: type v(5, k − 1) and all of B k−2 .…”

“…For example, let d o , d t ≥ 4, (d o , d t ) ̸ = (4, 4), and modify the procedure (CA) so that the result is a tiling by quadrilaterals in which every type-1 vertex has degree d o and every type-2 vertex degree d t . For a basepoint of given degree δ ≥ 4, the graph G(δ, d o , d t ) is a generalization of G(d, 4).…”

“…Limiting geometric properties of the graphs G(d, f ) have been studied before, as in [4][5][6]. Here we argue that asymptotic connectivity offers another sort of geometric measure, in the sense that the value 2 appears to indicate a boundary between hyperbolic and non-hyperbolic graphs.…”

Curvature, Geometry and Spectral Properties of Planar Graphs

“…For related results, see for instance [16,17,19,20]. The results (6) and (7) can be understood as hyperbolic properties of a graph.…”

“…Positivity of Cheeger's constant was also proven later by Higuchi [15] under the stronger assumption of negative vertex curvature in the case of tessellations. Explicit formulas for the Cheeger constant of regular tessellations can be found [16,17]. In [20] lower bounds for both types of Cheeger's constant are obtained in the context of locally tessellating graphs in terms of curvature.…”