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On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

Yilun Shang (2016)

Open Mathematics

As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition,...

On the rooted Tutte polynomial

F. Y. Wu, C. King, W. T. Lu (1999)

Annales de l'institut Fourier

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph.

On the unitary Cayley graph of a finite ring.

Akhtar, Reza, Boggess, Megan, Jackson-Henderson, Tiffany, Jiménez, Isidora, Karpman, Rachel, Kinzel, Amanda, Pritikin, Dan (2009)

The Electronic Journal of Combinatorics [electronic only]

Šachové úlohy v kombinatorice

Lucie Chybová (2018)

Pokroky matematiky, fyziky a astronomie

Článek pojednává o matematických úlohách souvisejících se šachovnicí a šachovými figurami. Ze šachu však budeme potřebovat pouze pravidla pro pohyb figur po šachovnici. Postupně se zaměřujeme na jezdcovy procházky po obdélníkových šachovnicích a dále na tzv. nezávislost a dominanci figur a vztah mezi nimi na čtvercových šachovnicích. Ukážeme, že některé problémy lze řešit elegantněji, pokud je přeformulujeme v řeči teorie grafů.

Skew-symmetric cluster algebras of finite mutation type

Anna Felikson, Michael Shapiro, Pavel Tumarkin (2012)

Journal of the European Mathematical Society

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices....

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