scholarly journals On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1374
Author(s):  
Umar Ali ◽  
Hassan Raza ◽  
Yasir Ahmed
Keyword(s):  
Structural Properties ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Molecular Graph ◽  
Decomposition Theorem ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Polynomial Decomposition ◽  
Number Of Spanning Trees

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

scholarly journals Spectra of Subdivision Vertex-Edge Join of Three Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 171 ◽  
Author(s):  
Fei Wen ◽  
You Zhang ◽  
Muchun Li
Keyword(s):  
Spanning Trees ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Graph Operation ◽  
Arbitrary Graphs ◽  
Number Of Spanning Trees

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

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 .

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 Spanning trees in random regular uniform hypergraphs

2021 ◽  
pp. 1-25
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Gary Liang
Keyword(s):  
Asymptotic Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Threshold Value ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Trivial Case ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Number Of Spanning Trees

Abstract Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950056
Author(s):  
Shreekant Patil ◽  
Mallikarjun Mathapati
Keyword(s):  
Spanning Trees ◽  
Product Formula ◽  
Regular Graphs ◽  
Kirchhoff Index ◽  
Cospectral Graphs ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Graph Operation ◽  
Product Of Graphs ◽  
Number Of Spanning Trees

Recently Indulal and Balakrishnan [Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13 (2016) 230–234] put forward a new graph operation, namely, the Indu–Bala product [Formula: see text] of graphs [Formula: see text] and [Formula: see text], and it is obtained from two disjoint copies of the join [Formula: see text] of [Formula: see text] and [Formula: see text] by joining the corresponding vertices in the two copies of [Formula: see text]. In this paper, we obtain the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of [Formula: see text] in terms of the corresponding spectra of [Formula: see text] and [Formula: see text]. As applications, these results enable us to construct infinitely many pairs of respective cospectral graphs. Further, the Laplacian spectra enable us to get the formulas of the number of spanning trees and Kirchhoff index of [Formula: see text] in terms of the Laplacian spectra of regular graphs [Formula: see text] and [Formula: see text].

scholarly journals The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Jia-Bao Liu ◽  
Qian Zheng ◽  
Sakander Hayat
Keyword(s):  
Structural Properties ◽  
Decomposition Theorem ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Triangular Matrices ◽  
Irregular Graphs ◽  
The Relationship ◽  
Explicit Expressions

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .

scholarly journals The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-10
Author(s):  
Jia-Bao Liu ◽  
Jing Chen ◽  
Jing Zhao ◽  
Shaohui Wang
Keyword(s):  
Spanning Trees ◽  
Characteristic Polynomials ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Number Of Spanning Trees ◽  
Structure Properties

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

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 Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

SPECTRAL ANALYSIS FOR WEIGHTED ITERATED TRIANGULATIONS OF GRAPHS

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850017 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Spectral Analysis ◽  
Closed Form ◽  
Structural Properties ◽  
Spanning Trees ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Successive Generations

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

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