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 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 Maximum Reciprocal Degree Resistance Distance Index of Unicyclic Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Gai-Xiang Cai ◽  
Xing-Xing Li ◽  
Gui-Dong Yu
Keyword(s):  
Connected Graph ◽  
Extremal Graph ◽  
Resistance Distance ◽  
Unicyclic Graphs

The reciprocal degree resistance distance index of a connected graph G is defined as RDRG=∑u,v⊆VGdGu+dGv/rGu,v, where rGu,v is the resistance distance between vertices u and v in G. Let Un denote the set of unicyclic graphs with n vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in Un and characterize the corresponding extremal graph.

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 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.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals The first two cacti with larger multiplicative eccentricity resistance-distance

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1783-1800
Author(s):  
Yunchao Hong ◽  
Zhongxun Zhu
Keyword(s):  
Connected Graph ◽  
Effective Resistance ◽  
Extremal Graphs ◽  
Resistance Distance ◽  
G Value

For a connected graph G, the multiplicative eccentricity resistance-distance ?*R(G) is defined as ?*R(G) = ?{x,y}?V(G)?(x)??(y)RG(x,y), where ?(?) is the eccentricity of the corresponding vertex and RG(x,y) is the effective resistance between vertices x and y. A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Let Cat(n;t) be the set of cacti possessing n vertices and t cycles, where 0 ? t ? n-1/2. In this paper, we first introduce some edge-grafting transformations which will increase ?*R(G). As their applications, the extremal graphs with maximum and second-maximum ?*R(G)-value in Cat(n,t) are characterized, respectively.

scholarly journals Resistance Distances in Linear Polyacene Graphs

2021 ◽  
Vol 8 ◽  
Author(s):  
Dayong Wang ◽  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Exact Expression ◽  
Effective Resistance ◽  
Electrical Network ◽  
Combinatorial Approach ◽  
Network Approach ◽  
Resistance Distance ◽  
Electric Network

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. In this article, using electric network approach and combinatorial approach, we derive exact expression for resistance distances between any two vertices of polyacene graphs.

scholarly journals Extensions of the Erdős–Gallai theorem and Luo’s theorem

2019 ◽  
Vol 29 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Bo Ning ◽  
Xing Peng
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Classical Theorem ◽  
Vertex Graph ◽  
Turán Number ◽  
Image Position

AbstractThe famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

scholarly journals Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

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