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.

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 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 Complexity of some of Pyramid Graphs Created from a Gear Graph

2018 ◽  
Author(s):  
Salama Daoud ◽  
Jia-Bao Liu
Keyword(s):  
Linear Algebra ◽  
Chebyshev Polynomials ◽  
Matrix Analysis ◽  
Analysis Techniques

In mathematics, one always aims to obtain new frameworks from specific ones. This also stratified to the regality of graphs, where one can produce numerous new graphs from a specific set of graphs. In this work we define some classes of pyramid graphs created by a gear graph and we derive straightforward formulas of the complexity of these graphs, using linear algebra matrix analysis techniques and employing knowledges of Chebyshev polynomials.

scholarly journals Complexity of Products of Some Complete and Complete Bipartite Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-25 ◽  
Author(s):  
S. N. Daoud
Keyword(s):  
Tensor Product ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Matrix Analysis ◽  
Normal Product ◽  
Analysis Techniques ◽  
Complete Bipartite Graphs ◽  
Composition Product ◽  
Number Of Spanning Trees

The number of spanning trees in graphs (networks) is an important invariant; it is also an important measure of reliability of a network. In this paper, we derive simple formulas of the complexity, number of spanning trees, of products of some complete and complete bipartite graphs such as cartesian product, normal product, composition product, tensor product, and symmetric product, using linear algebra and matrix analysis techniques.

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 Number of Spanning Trees in the Sequence of Some Graphs

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Jia-Bao Liu ◽  
S. N. Daoud
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Spanning Trees ◽  
Average Degree ◽  
Explicit Formulas ◽  
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 work, using knowledge of difference equations, we drive the explicit formulas for the number of spanning trees in the sequence of some graphs generated by a triangle by electrically equivalent transformations and rules of weighted generating function. Finally, we compare the entropy of our graphs with other studied graphs with average degree being 4, 5, and 6.

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 Comparing Graphs of Different Sizes

2017 ◽  
Vol 26 (5) ◽  
pp. 681-696
Author(s):  
RUSSELL LYONS
Keyword(s):  
Random Walks ◽  
Spanning Trees ◽  
Independent Sets ◽  
Finite Graph ◽  
Functions Of Operators ◽  
Number Of Spanning Trees

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

scholarly journals On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 314 ◽  
Author(s):  
Jia-Bao Liu ◽  
Jing Zhao ◽  
Zhongxun Zhu ◽  
Jinde Cao
Keyword(s):  
Spanning Trees ◽  
Decomposition Theorem ◽  
Laplacian Spectrum ◽  
Explicit Formulas ◽  
Kirchhoff Index ◽  
Regular Networks ◽  
Number Of Spanning Trees ◽  
Structure Properties

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n .

Sign in / Sign up

Export Citation Format

Share Document