The number of caterpillars

Harary, Frank; Schwenk, Allen J.

doi:10.1016/0012-365x(73)90067-8

Cited by 100 publications

(55 citation statements)

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“…where the numberĨ r+4 is also the number of all caterpillars with r + 4 (r ≥ 0) vertices, as showed Harary and Schwenk [22]. Recall that a caterpillar is a tree in which all vertices are within distance 1 of a central path.…”

Section: Lemma 1 Each Pair Of Adjacent Regular Pentagons In Plane (Smentioning

confidence: 97%

Pentagonal chains and annuli as models for designing nanostructures from cages

Rosenfeld

Dobrynin

Oliva³

et al. 2015

J Math Chem

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Carbon is the most versatile of chemical elements in combining with itself or other elements to form chains, rings, sheets, cages, and periodic 3D structures. One of the perspective trends for creating new molecules of nanotechnological interest deals with constructs which may be formed by chemically linking of cage molecules.The growing interest to fullerene polyhedra and other molecules with pentagonal rings raises also a question about geometrically consistent in E 3 nanoarchitectures which may be obtained by aggregating many such molecules. Simple examples are chains and rings assembled from pyramidal (car)borane subunits. Adequate geometrical models of such objects are a chain and an annulus built from regular pentagons wherein any two adjacent pentagons share an edge. Among arising combinatorial problems may be both analytical and constructive enumeration of such chains and annuli drawn in plane with no two edges crossing each other. This may also employ several mathematical disciplines, such as geometry, (spectral) graph theory, semigroup theory, theory of fractals, and others.We discuss some practical approaches for solving the mentioned mathematical problem.

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Section: Lemma 1 Each Pair Of Adjacent Regular Pentagons In Plane (Smentioning

confidence: 97%

Pentagonal chains and annuli as models for designing nanostructures from cages

Rosenfeld

Dobrynin

Oliva³

et al. 2015

J Math Chem

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show abstract

“…Harary and Schwenk (1973) used Pólya enumeration theory to show that the number of non-isomorphic caterpillars on n + 4 vertices is given by the elegant formula 2 n + 2 n/2 , and, in the same paper (p. 361), the authors credit A. Hobbs for introducing the term "caterpillar." Harary and Schwenk (1971) and Harary and Schwenk (1972) had earlier investigated the connectivity properties of graph powers of caterpillars.…”

Section: Connecting the Msc Mst And Tsp: Constants And Complexitymentioning

confidence: 99%

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

Jevtic

Steele

2015

Mathematics of OR

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Abstract. A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars (MSCs) where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton Hammersley theorem for the length of the shortest tour through a random sample.

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“…In C(P k ) all hairs are of length one. Harary and Schwenk [3], introduced Caterpillar graphs by saying: "Caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed". In chemical graph theory, caterpillar graphs are useful in studying topological properties of benzenoid hydrocarbons [2].…”

Section: Introductionmentioning

confidence: 99%

Maximal independent sets in a generalisation of caterpillar graph

Neethi¹,

Saxena²

2015

J Comb Optim

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A caterpillar graph is a tree which on removal of all its pendant vertices leaves a chordless path. The chordless path is called the backbone of the graph. The edges from the backbone to the pendant vertices are called the hairs of the caterpillar graph. Ortiz and Villanueva (C.Ortiz and M.Villanueva, Discrete Applied Mathematics, 160(3): 259-266, 2012) describe an algorithm, linear in the size of the output, for finding a family of maximal independent sets in a caterpillar graph.In this paper, we propose an algorithm, again linear in the output size, for a generalised caterpillar graph, where at each vertex of the backbone, there can be any number of hairs of length one and at most one hair of length two.

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The number of caterpillars

Cited by 100 publications

References 1 publication

Pentagonal chains and annuli as models for designing nanostructures from cages

Pentagonal chains and annuli as models for designing nanostructures from cages

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

Maximal independent sets in a generalisation of caterpillar graph

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