scholarly journals Maximizing the Number of Spanning Trees in a Connected Graph

2020 ◽  
Vol 66 (2) ◽  
pp. 1248-1260
Author(s):  
Huan Li ◽  
Stacy Patterson ◽  
Yuhao Yi ◽  
Zhongzhi Zhang
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Number Of Spanning Trees

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

On the Kirchhoff Index of Graphs

2013 ◽  
Vol 68 (8-9) ◽  
pp. 531-538 ◽  
Author(s):  
Kinkar C. Das
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Type Result ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Number Of Spanning Trees

Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ ... ≥ μn-1 > mn = 0. The Kirchhoff index of G is defined as [xxx] In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus-Gaddum-type result for the Kirchhoff index.

Trees, Trees, So Many Trees

2017 ◽  
Author(s):  
Allen J. Schwenk
Keyword(s):  
Connected Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Direct Argument ◽  
Acyclic Subgraph ◽  
Number Of Spanning Trees ◽  
Counting Trees

This chapter considers the problem of counting trees. Every connected graph G has a spanning tree, that is, a connected acyclic subgraph containing all the vertices of G. If G has no cycles, it is its own unique spanning tree. If G has cycles, we can locate any cycle and delete one of its edges. Repeat this process until no cycle remains. We have just constructed one of the spanning trees of G. Typically G will have many, many spanning trees. Let us use t(G) to denote the number of spanning trees in G. There are several ways to determine t(G). Some of these are direct argument, Kirchhoff's Matrix Tree Theorem, a variation of this theorem using eigenvalues, and Prüfer codes.

On the Minimal Graph with a Given Number of Spanning Trees

1970 ◽  
Vol 13 (4) ◽  
pp. 515-517 ◽  
Author(s):  
J. Sedláček
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Private Communication ◽  
Minimal Graph ◽  
Maximal Tree ◽  
Multiple Edges ◽  
Number Of Spanning Trees

Let G be a finite connected graph without loops or multiple edges. A maximal tree subgraph T of G is called a spanning tree of G. Denote by k(G) the number of all trees spanning the graph G. A. Rosa formulated the following problem (private communication): Let x(≠2) be a given positive integer and denote by α(x) the smallest positive integer y having the following property: There exists a graph G on y vertices with x spanning trees. Investigate the behavior of the function α(x).

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 $q(G)$).

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