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Enumeration of spanning trees in the sequence of Dürer graphs

100%

Li S.

Open Mathematics

2017

tom 15

nr 1

1591-1598

In this paper, we calculate the number of spanning trees in the sequence of Dürer graphs with a special feature that it has two alternate states. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and further derive relationships for spanning trees between the Dürer graphs and transformed graphs. By algebraic calculations, we obtain a closed-form formula for the number of spanning trees with regard to iteration step. Finally we compare the entropy of our graph with other studied graphs and see that its value of entropy lies in the interval of those of graphs with average degree being 3 and 4.

Gromov hyperbolicity of planar graphs

84%

Cantón A., Granados A., Pestana D., Rodríguez J.

Open Mathematics

2013

tom 11

nr 10

1817-1830

We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ2 with tiles which are parallelograms would be non-hyperbolic.

Ograniczanie wyników

2 Open Mathematics

1 Cantón A.

1 Granados A.

1 Li S.

1 Pestana D.

1 Rodríguez J.

1 2017

1 2013

Wyniki wyszukiwania

Enumeration of spanning trees in the sequence of Dürer graphs

Gromov hyperbolicity of planar graphs