scholarly journals A Breadth-first Search Tree Construction for Multiplicative Circulant Graphs

2021 ◽  
Vol 14 (1) ◽  
pp. 248-264
Author(s):  
John Rafael Macalisang Antalan ◽  
Francis Joseph Campena
Keyword(s):  
Spectral Radius ◽  
Average Distance ◽  
Wiener Index ◽  
Search Tree ◽  
Recursive Method ◽  
Breadth First Search ◽  
Circulant Graphs ◽  
Tree Construction

In this paper, we give a recursive method in constructing a breadth-first search tree for multiplicative circulant graphs of order power of odd. We then use the proposed construction in reproving some results concerning multiplicative circulant graph's diameter, average distance and distance spectral radius. We also determine the graph's Wiener index, vertex-forwarding index, and a bound for its edge-forwarding index. Finally, we discuss some possible research works in which the proposed construction can be applied.

An efficient distributed algorithm for constructing a breadth-first search tree

1989 ◽  
Vol 20 (10) ◽  
pp. 15-30 ◽  
Author(s):  
Jungho Park ◽  
Nobuki Tokura ◽  
Toshimitsu Masuzawa ◽  
Ken'Ichi Hagihara
Keyword(s):  
Distributed Algorithm ◽  
Search Tree ◽  
Breadth First Search

scholarly journals The Wiener Index of Circulant Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Houqing Zhou
Keyword(s):  
Distributed Computing ◽  
Interconnection Networks ◽  
Distance Matrix ◽  
Wiener Index ◽  
Important Class ◽  
Circulant Graphs ◽  
Parallel And Distributed Computing ◽  
Harary Index ◽  
Largest Eigenvalues ◽  
Reciprocal Distance Matrix

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. In this paper, we discuss the relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix and reciprocal distance matrix of circulants. We obtain the following consequence:W/λ=H/μ;2W/n=λ;2H/n=μ, whereW,Hdenote the Wiener index and the Harary index andλ,μdenote the largest eigenvalues of distance matrix and reciprocal distance matrix of circulant graphs, respectively. Moreover we also discuss the Wiener index of nonregular graphs with cut edges.

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

WIENER INDEX ON TRACEABLE AND HAMILTONIAN GRAPHS

2016 ◽  
Vol 94 (3) ◽  
pp. 362-372 ◽  
Author(s):  
RUIFANG LIU ◽  
XUE DU ◽  
HUICAI JIA
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Wiener Index ◽  
Hamiltonian Graphs

We give sufficient conditions for a graph to be traceable and Hamiltonian in terms of the Wiener index and the complement of the graph, which correct and extend the result of Yang [‘Wiener index and traceable graphs’, Bull. Aust. Math. Soc.88 (2013), 380–383]. We also present sufficient conditions for a bipartite graph to be traceable and Hamiltonian in terms of its Wiener index and quasicomplement. Finally, we give sufficient conditions for a graph or a bipartite graph to be traceable and Hamiltonian in terms of its distance spectral radius.

scholarly journals Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Shouliu Wei ◽  
Wai Chee Shiu ◽  
Xiaoling Ke ◽  
Jianwu Huang
Keyword(s):  
Difference Equation ◽  
Wiener Index ◽  
Recursive Method ◽  
Kirchhoff Index ◽  
Explicit Formulae ◽  
Expected Values ◽  
The Difference ◽  
Sum Of Distances

Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.

scholarly journals A survey of the all-pairs shortest paths problem and its variants in graphs

2016 ◽  
Vol 8 (1) ◽  
pp. 16-40 ◽  
Author(s):  
K. R. Udaya Kumar Reddy
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Average Distance ◽  
Shortest Paths ◽  
Wiener Index ◽  
Exact Results ◽  
Research Directions ◽  
All Pairs Shortest Paths ◽  
Static Version ◽  
Open Issues

Abstract There has been a great deal of interest in the computation of distances and shortest paths problem in graphs which is one of the central, and most studied, problems in (algorithmic) graph theory. In this paper, we survey the exact results of the static version of the all-pairs shortest paths problem and its variants namely, the Wiener index, the average distance, and the minimum average distance spanning tree (MAD tree in short) in graphs (focusing mainly on algorithmic results for such problems). Along the way we also mention some important open issues and further research directions in these areas.

ON THE TOTAL DISTANCE AND DIAMETER OF GRAPHS

2018 ◽  
Vol 98 (1) ◽  
pp. 14-17
Author(s):  
HONGBO HUA
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Average Distance ◽  
Wiener Index ◽  
Connected Graphs ◽  
Total Distance

The total distance (or Wiener index) of a connected graph$G$is the sum of all distances between unordered pairs of vertices of$G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’,Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if$G$has diameter$D>2$and order$2D+1$, then the total distance of$G$is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

scholarly journals Graphs with Given Degree Sequence and Maximal Spectral Radius

10.37236/843 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Türker Bıyıkoğlu ◽  
Josef Leydold
Keyword(s):  
Spectral Radius ◽  
Degree Sequence ◽  
Breadth First Search ◽  
Connected Graphs

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization.

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