A simple formula for the number of spanning trees of line graphs

2017 ◽  
Vol 88 (2) ◽  
pp. 294-301 ◽  
Author(s):  
Helin Gong ◽  
Xian'an Jin
Keyword(s):  
Simple Formula ◽  
Spanning Trees ◽  
Line Graphs ◽  
Number Of Spanning Trees

scholarly journals The Number of Spanning Trees of the Cartesian Product of Regular Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Mei-Hui Wu ◽  
Long-Yeu Chung
Keyword(s):  
Simple Formula ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Regular Graphs ◽  
Electrical Circuits ◽  
Important Measure ◽  
Computational Perspective ◽  
Regular Networks ◽  
Number Of Spanning Trees

The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.

scholarly journals Properties of Commuting Graphs over Semidihedral Groups

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 103
Author(s):  
Tao Cheng ◽  
Matthias Dehmer ◽  
Frank Emmert-Streib ◽  
Yongtao Li ◽  
Weijun Liu
Keyword(s):  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Vertex Connectivity ◽  
Laplacian Energy ◽  
Number Of Spanning Trees

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

scholarly journals On the characterization of graphs with maximum number of spanning trees

1998 ◽  
Vol 179 (1-3) ◽  
pp. 155-166 ◽  
Author(s):  
L. Petingi ◽  
F. Boesch ◽  
C. Suffel
Keyword(s):  
Spanning Trees ◽  
Number Of Spanning Trees

Determinant of links, spanning trees, and a theorem of Shank

2016 ◽  
Vol 25 (09) ◽  
pp. 1641005
Author(s):  
Jun Ge ◽  
Lianzhu Zhang
Keyword(s):  
Spanning Trees ◽  
Elementary Proof ◽  
Number Of Spanning Trees

In this note, we first give an alternative elementary proof of the relation between the determinant of a link and the spanning trees of the corresponding Tait graph. Then, we use this relation to give an extremely short, knot theoretical proof of a theorem due to Shank stating that a link has component number one if and only if the number of spanning trees of its Tait graph is odd.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

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 $c(C(n))$.

Sign in / Sign up

Export Citation Format

Share Document