Algorithms for Computing Wiener Indices of Acyclic and Unicyclic Graphs

Complexity ◽

10.1155/2021/6663306 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Bo Bi ◽

Muhammad Kamran Jamil ◽

Khawaja Muhammad Fahd ◽

Tian-Le Sun ◽

Imran Ahmad ◽

...

Keyword(s):

Physical Properties ◽

Topological Index ◽

Linear Time ◽

Molecular Graph ◽

Wiener Index ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Unicyclic Graphs ◽

Polarity Index ◽

Acyclic Graphs

Let G = V G , E G be a molecular graph, where V G and E G are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity, and the terminal Wiener indices are the distance-based topological indices. In this paper, we described a linear time algorithm (LTA) that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and the Wiener polarity index. All these algorithms compute the indices in time O n .

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Terminal Wiener Index of Fibonacci trees

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.b4542.029320 ◽

2020 ◽

Vol 9 (3) ◽

pp. 2533-2535

Keyword(s):

Closed Form ◽

Linear Time ◽

Wiener Index ◽

Time Algorithm ◽

Closed Form Expression ◽

Linear Time Algorithm ◽

Form Expression ◽

Sum Of Distances

The terminal Wiener index of a tree is defined as the sum of distances between all leaf pairs of T. We derive closed form expression for the terminal Wiener index of fibonacci trees. We also describe a linear time algorithm to compute terminal Wiener index of a tree.

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Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

Trends in Computational and Applied Mathematics ◽

10.5540/tcam.2021.022.04.00659 ◽

2021 ◽

Vol 22 (4) ◽

pp. 659-674

Author(s):

R. O. Braga ◽

V. M. Rodrigues ◽

R. O. Silva

Keyword(s):

Linear Time ◽

Symmetric Matrix ◽

Time Algorithm ◽

Real Interval ◽

Linear Time Algorithm ◽

Matrix Representations ◽

Unicyclic Graphs ◽

Laplacian Matrices ◽

Underlying Graph ◽

Signless Laplacian

We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2021.07.002 ◽

2021 ◽

Author(s):

Zdeněk Dvořák ◽

Daniel Král' ◽

Robin Thomas

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Graphs On Surfaces

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Nearly Linear Time Algorithm for Mean Hitting Times of Random Walks on a Graph

Proceedings of the 13th International Conference on Web Search and Data Mining ◽

10.1145/3336191.3371777 ◽

2020 ◽

Author(s):

Zuobai Zhang ◽

Wanyue Xu ◽

Zhongzhi Zhang

Keyword(s):

Random Walks ◽

Linear Time ◽

Time Algorithm ◽

Hitting Times ◽

Linear Time Algorithm

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A Linear time algorithm for deciding security

10.1109/sfcs.1976.1 ◽

1976 ◽

Cited By ~ 33

Author(s):

A. K. Jones ◽

R. J. Lipton ◽

L. Snyder

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm

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AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

International Journal of Foundations of Computer Science ◽

10.1142/s0129054100000247 ◽

2000 ◽

Vol 11 (03) ◽

pp. 365-371 ◽

Cited By ~ 33

Author(s):

LJUBOMIR PERKOVIĆ ◽

BRUCE REED

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Tree Decomposition ◽

Linear Time Algorithm ◽

Input Graph ◽

Primary Motivation ◽

Small Width ◽

Tree Decompositions ◽

Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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