scholarly journals The Number of Spanning Trees in the Composition Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Feng Li
Keyword(s):  
Linear Algebra ◽  
Spanning Trees ◽  
Matrix Theory ◽  
Structural Parameters ◽  
The Other ◽  
Arbitrary Graph ◽  
Large Graphs ◽  
Laplacian Eigenvalues ◽  
Number Of Spanning Trees

Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of vertices and eigenvalues of the small graphs only.

scholarly journals Complexity of Some Generalized Operations on Networks

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-29
Author(s):  
Muhammad Javaid ◽  
Hafiz Usman Afzal ◽  
Shaohui Wang
Keyword(s):  
Linear Algebra ◽  
Chebyshev Polynomials ◽  
Spanning Trees ◽  
Matrix Theory ◽  
Connected Components ◽  
Number Of Spanning Trees

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of S n ), sum ( S n + W 3 , S n + K 2 , and C n ∘ K 2 + K 1 ), product ( S n ⊠ K 2 and W n ∘ K 2 ), semitotal networks ( Q S n and R S n ), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.

scholarly journals Number of Spanning Trees of Different Products of Complete and Complete Bipartite Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-23 ◽  
Author(s):  
S. N. Daoud
Keyword(s):  
Tensor Product ◽  
Linear Algebra ◽  
Spanning Trees ◽  
Matrix Theory ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Normal Product ◽  
Complete Bipartite Graphs ◽  
Composition Product ◽  
Number Of Spanning Trees

Spanning trees have been found to be structures of paramount importance in both theoretical and practical problems. In this paper we derive new formulas for the complexity, number of spanning trees, of some products of complete and complete bipartite graphs such as Cartesian product, normal product, composition product, tensor product, symmetric product, and strong sum, using linear algebra and matrix theory techniques.

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

2016 ◽  
Vol 14 (1) ◽  
pp. 641-648 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Spanning Trees ◽  
Line Graph ◽  
Upper And Lower Bounds ◽  
Barycentric Subdivision ◽  
Electrical Networks ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Vertex Degrees ◽  
Number Of Spanning Trees

Abstract 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, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(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.

scholarly journals The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Raihana Mokhlissi ◽  
Dounia Lotfi ◽  
Joyati Debnath ◽  
Mohamed El Marraki ◽  
Noussaima EL Khattabi
Keyword(s):  
Complex Networks ◽  
Spanning Tree ◽  
Decomposition Method ◽  
Spanning Trees ◽  
Small World ◽  
Theoretical Computer Science ◽  
The Other ◽  
Small World Networks ◽  
Theoretical Computer ◽  
Number Of Spanning Trees

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

scholarly journals The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 689 ◽  
Author(s):  
Jia-Bao Liu ◽  
Salama Nagy Daoud
Keyword(s):  
Linear Algebra ◽  
Chebyshev Polynomials ◽  
Spanning Trees ◽  
Finite Graph ◽  
Numerical Results ◽  
Matrix Analysis ◽  
Explicit Formulas ◽  
Analysis Techniques ◽  
Number Of Spanning Trees ◽  
Mathematics And Physics

The methods of measuring the complexity (spanning trees) in a finite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we defined some classes of pyramid graphs created by a gear graph then we developed the Kirchhoff's matrix tree theorem method to produce explicit formulas for the complexity of these graphs, using linear algebra, matrix analysis techniques, and employing knowledge of Chebyshev polynomials. Finally, we gave some numerical results for the number of spanning trees of the studied graphs.

Study on Applications of Laplacian Spectra for a Network

2013 ◽  
Vol 753-755 ◽  
pp. 2859-2862
Author(s):  
Hai Tang Wang
Keyword(s):  
Recurrence Relations ◽  
Spanning Trees ◽  
Small World ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Scale Free ◽  
Structure And Dynamics ◽  
Laplacian Spectra ◽  
Number Of Spanning Trees ◽  
Biological Field

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.

scholarly journals On the Complexity of a Class of Pyramid Graphs and Chebyshev Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
S. N. Daoud
Keyword(s):  
Linear Algebra ◽  
Chebyshev Polynomials ◽  
Spanning Trees ◽  
Matrix Analysis ◽  
Analysis Techniques ◽  
Number Of Spanning Trees

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this paper we define a class of pyramid graphs and derive simple formulas of the complexity, number of spanning trees, of these graphs, using linear algebra, Chebyshev polynomials, and matrix analysis techniques.

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.

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