scholarly journals A FORMULA FOR THE NUMBER OF SPANNING TREES IN CIRCULANT GRAPHS WITH NONFIXED GENERATORS AND DISCRETE TORI

2015 ◽  
Vol 92 (3) ◽  
pp. 365-373 ◽  
Author(s):  
JUSTINE LOUIS
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Closed Formula ◽  
Circulant Graphs ◽  
The Matrix ◽  
Number Of Spanning Trees

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

scholarly journals A Positive Combinatorial Formula for the Complexity of the $q$-Analog of the $n$-Cube

10.37236/2389 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Murali Krishna Srinivasan
Keyword(s):  
Graph Theory ◽  
Explicit Formula ◽  
Classical Result ◽  
Spanning Trees ◽  
Combinatorial Formula ◽  
Algebraic Graph Theory ◽  
The Matrix ◽  
Number Of Spanning Trees

The number of spanning trees of a graph $G$ is called the complexity of $G$. A classical result in algebraic graph theory explicitly diagonalizes the Laplacian of the $n$-cube $C(n)$  and yields, using the Matrix-Tree theorem, an explicit formula for $c(C(n))$. In this paper we explicitly block diagonalize the Laplacian of the $q$-analog $C_q(n)$ of $C(n)$ and use this, along with the Matrix-Tree theorem, to give a positive combinatorial formula for $c(C_q(n))$. We also explain how setting $q=1$ in the formula for $c(C_q(n))$ recovers the formula for $c(C(n))$.

scholarly journals On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

2020 ◽  
Vol 27 (01) ◽  
pp. 87-94
Author(s):  
A.D. Mednykh ◽  
I.A. Mednykh
Keyword(s):  
Rational Function ◽  
Generating Function ◽  
Spanning Trees ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Integer Coefficients ◽  
Number Formula ◽  
Number Of Spanning Trees

Let [Formula: see text] be the generating function for the number [Formula: see text] of spanning trees in the circulant graph Cn(s1, s2, …, sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, …, sk, n) of odd valency. We illustrate the obtained results by a series of examples.

scholarly journals Further analysis of the number of spanning trees in circulant graphs

2006 ◽  
Vol 306 (22) ◽  
pp. 2817-2827 ◽  
Author(s):  
Talip Atajan ◽  
Xuerong Yong ◽  
Hiroshi Inaba
Keyword(s):  
Spanning Trees ◽  
Circulant Graphs ◽  
Number Of Spanning Trees

scholarly journals The number of spanning trees in circulant graphs

2000 ◽  
Vol 223 (1-3) ◽  
pp. 337-350 ◽  
Author(s):  
Yuanping Zhang ◽  
Xuerong Yong ◽  
Mordecai J. Golin
Keyword(s):  
Spanning Trees ◽  
Circulant Graphs ◽  
Number Of Spanning Trees

On the number of spanning trees in directed circulant graphs

Networks ◽  
10.1002/net.2 ◽  
2001 ◽  
Vol 37 (3) ◽  
pp. 129-133 ◽  
Author(s):  
Zbigniew Lonc ◽  
Krzysztof Parol ◽  
Jacek M. Wojciechowski
Keyword(s):  
Spanning Trees ◽  
Circulant Graphs ◽  
Number Of Spanning Trees

scholarly journals On the Spanning Trees of the Hypercube and other Products of Graphs

10.37236/2510 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Olivier Bernardi
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Product Formula ◽  
General Context ◽  
Complete Graphs ◽  
Independence Property ◽  
Number Of Spanning Trees ◽  
Tree Approach

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

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