scholarly journals The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs

2016 ◽  
Vol 177 ◽  
pp. 441-446 ◽  
Author(s):  
Qun Liu ◽  
Jia-Bao Liu ◽  
Jinde Cao
Keyword(s):  
Kirchhoff Index

NetKI: A Kirchhoff Index Based Statistical Graph Embedding in Nearly Linear Time

2020 ◽  
Author(s):  
Anwar Said ◽  
Saeed-Ul Hassan ◽  
Waseem Abbas ◽  
Mudassir Shabbir
Keyword(s):  
Linear Time ◽  
Graph Embedding ◽  
Kirchhoff Index ◽  
Statistical Graph

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 Kirchhoff index and degree Kirchhoff index of complete multipartite graphs

2017 ◽  
Vol 232 ◽  
pp. 41-49 ◽  
Author(s):  
Ravindra B. Bapat ◽  
Masoud Karimi ◽  
Jia-Bao Liu
Keyword(s):  
Kirchhoff Index ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite

A Note on the Kirchhoff and Additive Degree-Kirchhoff Indices of Graphs

2015 ◽  
Vol 70 (6) ◽  
pp. 459-463 ◽  
Author(s):  
Yujun Yang ◽  
Douglas J. Klein
Keyword(s):  
Upper Bound ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Resistance Distance

AbstractTwo resistance-distance-based graph invariants, namely, the Kirchhoff index and the additive degree-Kirchhoff index, are studied. A relation between them is established, with inequalities for the additive degree-Kirchhoff index arising via the Kirchhoff index along with minimum, maximum, and average degrees. Bounds for the Kirchhoff and additive degree-Kirchhoff indices are also determined, and extremal graphs are characterised. In addition, an upper bound for the additive degree-Kirchhoff index is established to improve a previously known result.

scholarly journals Mean Hitting Time for Random Walks on a Class of Sparse Networks

Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 34
Author(s):  
Jing Su ◽  
Xiaomin Wang ◽  
Bing Yao
Keyword(s):  
Random Walks ◽  
Complete Graph ◽  
Hitting Time ◽  
Electrical Networks ◽  
Kirchhoff Index ◽  
Scale Free ◽  
Closed Form Solutions ◽  
Sparse Networks ◽  
Similar Dynamic ◽  
Recursive Relations

For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks G(t) in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that G(t) has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of G(t), and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.

scholarly journals The expected values for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random cyclooctatetraene chain

2021 ◽  
Author(s):  
Xianya Geng ◽  
Jinfeng Qi ◽  
Minjie Zhang
Keyword(s):  
Kirchhoff Index ◽  
Schultz Index ◽  
Expected Values ◽  
Gutman Index ◽  
Analytical Expressions

In this paper, we mainly solve the explicit analytical expressions for the expected values of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random cyclooctatetraene chain with $n$ octagons. We also obtain the average values of these four indices with respect to the set of all these cyclooctatetraene chains.

scholarly journals Resistance Distance and Kirchhoff Index of Graphs with Pockets

2018 ◽  
Author(s):  
Qun Liu ◽  
Jiabao Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

scholarly journals Group inverse matrix of the normalized Laplacian on subdivision networks

2020 ◽  
pp. 23-23
Author(s):  
Ángeles Carmona ◽  
Margarida Mitjana ◽  
Enric Monsó
Keyword(s):  
Inverse Matrix ◽  
Group Inverse ◽  
Kirchhoff Index ◽  
Base Network

In this paper we consider a subdivision of a given network and we show how the group inverse matrix of the normalized laplacian of the subdivision network is related to the group inverse matrix of the normalized laplacian of the initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both structures and takes advantage on the properties of the group inverse matrix. As a consequence we get formulae for effective resistances and the Kirchhoff Index of the subdivision network expressed in terms of its corresponding in the base network. Finally, we study two examples where the base network are the star and the wheel, respectively.

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.

On the Kirchhoff index of some toroidal lattices

2011 ◽  
Vol 59 (6) ◽  
pp. 645-650 ◽  
Author(s):  
Luzhen Ye
Keyword(s):  
Kirchhoff Index
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