Subgraph statistics in subcritical graph classes

Abstract

Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft

Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs

“…The third and fourth correspond to the symbolic constructions: Z C • = Z SET(B • • C • ) and G = SET(C). The former relation is well-known (see for instance [14, p.10], [12], [8]) and is based on the decomposition of a graph into 2-connected components. The latter one is straightforward.…”

“…In order to pass to labelled 2-connected, connected, and then general outerplanar graphs, we use the multivariate analogues of the equations in Table 2 This has indeed a minimal positive solution (τ, z 0 ), since the outerplanar graph class belongs to the subcritical family of graphs [8], i.e. z 0 C (z 0 , 1) < r(1), where z 0 is the radius of convergence of C (z, 1) and r(1) the one of B (z, 1).…”

“…In the case of outerplanar graphs, the limit laws are expected from [8]. However, in [8] there is no constructive way to compute the relevant constants. This is a contribution of this work, that offers specific defining equations for the function B .…”

“…Figure 4: The composite roots when 5-cycles are avoided. The dissection 4 [8] is the only maximal one among them.…”

Threshold functions for small subgraphs: an analytic approach

“…The variables z and w are redundant in the generating function F (z, w, y), because each vertex has a degree, and the sum of the degrees is twice the number of edges. Thus, given two variables a, b, we have formally F (az, bw, y) = F z, w, ab d/2 y d d≥0 (15) Notice that the sum of degrees being even, it ensures that the exponents of b are non-negative integers, defining a proper formal power series. Given a multigraph family F with generating function F (z, w, y) (with the convention stated in Equation (14)), the total weight MG…”