scholarly journals Spectra of some interesting combinatorial matrices related to oriented spanning trees on a directed graph

1996 ◽  
Vol 5 (1) ◽  
Author(s):  
ChristosA. Athanasiadis
Keyword(s):  
Directed Graph ◽  
Spanning Trees

scholarly journals A Combinatorial Proof of a Formula of Biane and Chapuy

10.37236/7061 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sinho Chewi ◽  
Venkat Anantharam
Keyword(s):  
Directed Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Outgoing Edge ◽  
Alternative Proof ◽  
Combinatorial Proof ◽  
Tree Graph ◽  
Weighted Directed Graph ◽  
Strongly Connected

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.

scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

scholarly journals An Algorithm for Enumerating All Spanning Trees of a Directed Graph

Algorithmica ◽  
2000 ◽  
Vol 27 (2) ◽  
pp. 120-130 ◽  
Author(s):  
S. Kapoor ◽  
H. Ramesh
Keyword(s):  
Directed Graph ◽  
Spanning Trees
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