Degree Kirchhoff Index of Bicyclic Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Zikai Tang ◽  
Hanyuan Deng
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Maximum And Minimum Degree ◽  
Bicyclic Graphs

AbstractLet G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

scholarly journals The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 83
Author(s):  
Fangguo He ◽  
Zhongxun Zhu
Keyword(s):  
Effective Resistance ◽  
Extremal Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set

For a graph G, the resistance distance r G ( x , y ) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ ( G ) = ∑ { x , y } ⊂ V ( G ) d G ( x ) d G ( y ) r G ( x , y ) , where d G ( x ) is the degree of vertex x, and V ( G ) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t ( n ; t ) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ ( G ) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

scholarly journals Bicyclic graphs with maximum degree resistance distance

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1625-1632 ◽  
Author(s):  
Junfeng Du ◽  
Jianhua Tu
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Theoretical Chemistry ◽  
Graph Invariants ◽  
Resistance Distance ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Bicyclic Graphs

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. Recently, Gutman, Feng and Yu (Transactions on Combinatorics, 01 (2012) 27- 40) introduced the degree resistance distance of a graph G, which is defined as DR(G) = ?{u,v}?V(G)[dG(u)+dG(v)]RG(u,v), where dG(u) is the degree of vertex u of the graph G, and RG(u, v) denotes the resistance distance between the vertices u and v of the graph G. Further, they characterized n-vertex unicyclic graphs having minimum and second minimum degree resistance distance. In this paper, we characterize n-vertex bicyclic graphs having maximum degree resistance distance.

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 Maximum Reciprocal Degree Resistance Distance Index of Bicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Gaixiang Cai ◽  
Xing-Xing Li ◽  
Guidong Yu
Keyword(s):  
Connected Graph ◽  
Extremal Graph ◽  
Resistance Distance ◽  
Bicyclic Graphs

The reciprocal degree resistance distance index of a connected graph G is defined as RDR G = ∑ u , v ⊆ V G d G u + d G v / r G u , v , where r G u , v is the resistance distance between vertices u and v in G . Let ℬ n denote the set of bicyclic graphs without common edges and with n vertices. We study the graph with the maximum reciprocal degree resistance distance index among all graphs in ℬ n and characterize the corresponding extremal graph.

scholarly journals Some Bounds for the Kirchhoff Index of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Electrical Network ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Fullerene Graphs

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

THE GUTMAN INDEX OF UNICYCLIC GRAPHS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250031 ◽  
Author(s):  
LIHUA FENG
Keyword(s):  
Connected Graph ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Gutman Index

Let G be a connected graph with vertex set V(G). The Gutman index of G is defined as S(G) = ∑{u, v}⊆V(G) d(u)d(v)d(u, v), where d(u) is the degree of vertex u, and d(u, v) denotes the distance between u and v. In this paper, we characterize n-vertex unicyclic graphs with girth k, having minimal Gutman index.

scholarly journals Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

2018 ◽  
Vol 36 (2) ◽  
pp. 9-15
Author(s):  
Vida Ahmadi ◽  
Mohammad Reza Darafshe
Keyword(s):  
Connected Graph ◽  
Zagreb Indices ◽  
The Third ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Simple Connected Graph

Let  be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: ,   and   where  is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph  are defind by:  and . The first and second Zagreb coindices of graph are defind by:  and .  and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some  graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

scholarly journals DEGREE DISTANCE AND MINIMUM DEGREE

2012 ◽  
Vol 87 (2) ◽  
pp. 255-271 ◽  
Author(s):  
S. MUKWEMBI ◽  
S. MUNYIRA
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Degree Of Vertex ◽  
Degree Distance ◽  
Image Position ◽  
Appl Math ◽  
Extremal Properties

AbstractLet G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u+deg v) d(u,v), where deg w is the degree of vertex w and d(u,v) denotes the distance between u and v. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that \[ D^\prime (G)\le \frac {1}{4}\,dn\biggl (n-\frac {d}{3}(\delta +1)\biggr )^2+O(n^3). \] As a corollary, we obtain the bound D′ (G)≤n4 /(9(δ+1) )+O(n3) for a graph G of order n and minimum degree δ. This result, apart from improving on a result of Dankelmann et al. [‘On the degree distance of a graph’, Discrete Appl. Math.157 (2009), 2773–2777] for graphs of given order and minimum degree, completely settles a conjecture of Tomescu [‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math.98(1999), 159–163].

scholarly journals Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wenjie Ning ◽  
Kun Wang ◽  
Hassan Raza
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Effective Resistance ◽  
Resistance Distance ◽  
Bicyclic Graph ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.

scholarly journals Resistance Distance and Kirchhoff Index for a Class of Graphs

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
WanJun Yin ◽  
ZhengFeng Ming ◽  
Qun Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk,Hv] 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, Hv is a simple graph of order m≥2, and 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, Huv is a simple graph of order m≥3, and uv is a specified edge of Huv such 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.

Sign in / Sign up

Export Citation Format

Share Document