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 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 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 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 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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