scholarly journals The Kirchhoff Index of Some Combinatorial Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao ◽  
Fu-Tao Hu
Keyword(s):  
Characteristic Polynomial ◽  
Line Graph ◽  
Laplacian Matrix ◽  
Effective Resistance ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Explicit Formulae ◽  
Folded Hypercube

The Kirchhoff index Kf(G) is the sum of the effective resistance distances between all pairs of vertices inG. The hypercubeQnand the folded hypercubeFQnare well known networks due to their perfect properties. The graphG∗, constructed fromG, is the line graph of the subdivision graphS(G). In this paper, explicit formulae expressing the Kirchhoff index of(Qn)∗and(FQn)∗are found by deducing the characteristic polynomial of the Laplacian matrix ofG∗in terms of that ofG.

scholarly journals The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jiabao Liu ◽  
Xiang-Feng Pan ◽  
Yi Wang ◽  
Jinde Cao
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube

Then-dimensional folded hypercubeFQnis an important and attractive variant of then-dimensional hypercubeQn, which is obtained fromQnby adding an edge between any pair of vertices complementary edges.FQnis superior toQnin many measurements, such as the diameter ofFQnwhich is⌈n/2⌉, about a half of the diameter in terms ofQn. The Kirchhoff indexKf(G)is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the folded hypercubes networksFQnand its three variant networksl(FQn),s(FQn), andt(FQn)on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofFQn,l(FQn),s(FQn), andt(FQn)were proposed, respectively.

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 On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
S. R. Jog ◽  
Raju Kotambari
Keyword(s):  
Line Graph ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Complete Graphs ◽  
Laplacian Spectrum ◽  
Spectral Graph ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Energy ◽  
Signless Laplacian

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

scholarly journals Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Harishchandra S. Ramane ◽  
Shaila B. Gudimani ◽  
Sumedha S. Shinde
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Induced Subgraphs ◽  
Subdivision Graph ◽  
Signless Laplacian ◽  
The Matrix ◽  
Derivatives Of ◽  
Degree Matrix

The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

2017 ◽  
Vol 72 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Haiyan Chen
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Effective Resistance ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Join Graph ◽  
Explicit Formulae

AbstractThe resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

scholarly journals Hyperbolicity on Graph Operators

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 360 ◽  
Author(s):  
J. Méndez-Bermúdez ◽  
Rosalío Reyes ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Line Graph ◽  
Discrete Mathematics ◽  
Total Graph ◽  
Subdivision Graph ◽  
Gromov Product

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

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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