The limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain

2021 ◽  
Vol 299 ◽  
pp. 62-73
Author(s):  
Jinlian Zhang ◽  
Xuhui Peng ◽  
Hanlin Chen
Keyword(s):  
Kirchhoff Index ◽  
Schultz Index ◽  
Gutman Index

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.

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

2020 ◽  
Vol 282 ◽  
pp. 243-256 ◽  
Author(s):  
Leilei Zhang ◽  
Qishun Li ◽  
Shuchao Li ◽  
Minjie Zhang
Keyword(s):  
Kirchhoff Index ◽  
Schultz Index ◽  
Expected Values ◽  
Gutman 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

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

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 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 Computation of Gutman index of some cactus chains

2018 ◽  
Vol 6 (1) ◽  
pp. 138
Author(s):  
Ali Sadeghieh ◽  
Nima Ghanbari ◽  
Saeid Alikhani
Keyword(s):  
Gutman Index
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