# Exact Largest and Smallest Size of Components

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Daniel Panario, Jun 30, 2015 Available from:- References (27)
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**ABSTRACT:**This paper begins with the observation that half of all graphs containing no induced path of length 3 are disconnected. We generalize this in several directions. First, we give necessary and sufficient conditions (in terms of generating functions) for the probability of connectedness in a suitable class of graphs to tend to a limit strictly between zero and one. Next we give a general framework in which this and related questions can be posed, involving operations on classes of finite structures. Finally, we discuss briefly an algebra associated with such a class of structures, and give a conjecture about its structure. 1 1 Introduction The class of graphs containing no induced path of length 3 has many remarkable properties, stemming from the following well-known observation. Recall that an induced subgraph of a graph consists of a subset S of the vertex set together with all edges contained in S. Proposition 1.1 Let G be a finite graph with more than one vertex, containin...Combinatorics Probability and Computing 06/1998; 8(1-2). DOI:10.1017/S0963548398003423 · 0.62 Impact Factor -
##### Article: Random Mapping Statistics

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**ABSTRACT:**Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed. - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n 3/2 ) , the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices .) For sparse tables, the construction cost has expectation O(n) , standard deviation O ( ), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.Algorithmica 12/1998; 22:490-515. DOI:10.1007/PL00009236 · 0.57 Impact Factor