Spanning tree formulas and chebyshev polynomials

Boesch, Francis T.; Prodinger, Helmut

doi:10.1007/bf01788093

Cited by 81 publications

(88 citation statements)

References 9 publications

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“…Myers [20] proved this using identities involving weighted compositions; the proof by Benjamin and Yerger [2] is based on counting imperfect matchings. A useful tool for solving such problems is Chebyshev polynomials, see [17,3,25]. Our proof of the identity f (C n ) = n + n+1 − 2 presented here for completeness is based on relations between forests found before.…”

Section: Spanning Rooted Forests In Paths Cycles and T-caterpillarsmentioning

confidence: 97%

Spanning forests and the golden ratio

Chebotarev

2008

Discrete Applied Mathematics

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For a graph G, let f ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f ij 's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F = (f ij ) n×n /f , where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F −1 = I + L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to −1 or to 1 − −1 , where is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as "golden introverts" and "golden extroverts," respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.

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Section: Spanning Rooted Forests In Paths Cycles and T-caterpillarsmentioning

confidence: 97%

Spanning forests and the golden ratio

Chebotarev

2008

Discrete Applied Mathematics

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“…For a few special kinds of undirected circulant graphs, explicit formulas for the number of spanning trees were found (see Baron et al [1], Boesch and Prodinger [2], Sedlaček [8,9]). …”

Section: Introductionmentioning

confidence: 99%

On the number of spanning trees in directed circulant graphs

2001

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“…As a result, the number of spanning trees of some specific families of graphs can be given explicitly with reference to certain graph parameters. For example, the closed formula for counting the number of spanning trees of graphs, including complete graphs, the triangle graphs, the Möbius laders, the complete multipartite graphs, and the "almost-complete" graphs, can be referred in [10,11]. Recently, the number of spanning trees of some graphs, for example, the circulant graphs, the square of a cycle, the threshold graphs, some multicomplete/star-related graphs, and spanning trees with few leaves in weighted graphs can also be obtained [6,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

The number of spanning trees in a new lexicographic product of graphs

Liang

Zhao

2014

Sci. China Inf. Sci.

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The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.

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Spanning tree formulas and chebyshev polynomials

Cited by 81 publications

References 9 publications

Spanning forests and the golden ratio

Spanning forests and the golden ratio

On the number of spanning trees in directed circulant graphs

The number of spanning trees in a new lexicographic product of graphs

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