scholarly journals On Harmonic Index and Diameter of Quasi-Tree Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
A. Abdolghafourian ◽  
Mohammad A. Iranmanesh
Keyword(s):  
Lower Bounds ◽  
Tree Graph ◽  
Harmonic Index ◽  
Tree Graphs ◽  
Degree Of A Vertex

The harmonic index of a graph G ( H G ) is defined as the sum of the weights 2 / d u + d v for all edges u v of G , where d u is the degree of a vertex u in G . In this paper, we show that H G ≥ D G + 5 / 3 − n / 2 and H G ≥ 1 / 2 + 2 / 3 n − 2 D G , where G is a quasi-tree graph of order n and diameter D G . Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.

Sharp lower bounds on the sum-connectivity index of quasi-tree graphs

2021 ◽  
pp. 2150136
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Tree Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. A graph [Formula: see text] is called quasi-tree, if there exists [Formula: see text] such that [Formula: see text] is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs.

scholarly journals A lower bound for the harmonic index of a graph with minimum degree at least three

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2249-2260 ◽  
Author(s):  
Minghong Cheng ◽  
Ligong Wang
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Harmonic Index ◽  
Degree Of A Vertex

The harmonic index H(G) of a graph G is the sum of the weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this work, a lower bound for the harmonic index of a graph with minimum degree at least three is obtained and the corresponding extremal graph is characterized.

General sum-connectivity index of unicyclic graphs with given diameter and girth

2021 ◽  
pp. 2150140
Author(s):  
Tomáš Vetrík
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Connectivity Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Harmonic Index

Topological indices of graphs have been studied due to their extensive applications in chemistry. We obtain lower bounds on the general sum-connectivity index [Formula: see text] for unicyclic graphs [Formula: see text] of given girth and diameter, and for unicyclic graphs of given diameter, where [Formula: see text]. We present the extremal graphs for all the bounds. Our results generalize previously known results on the harmonic index for unicyclic graphs of given diameter.

scholarly journals RAG-Web: RNA structure prediction/design using RNA-As-Graphs

2019 ◽  
Author(s):  
Grace Meng ◽  
Marva Tariq ◽  
Swati Jain ◽  
Shereef Elmetwaly ◽  
Tamar Schlick
Keyword(s):  
Secondary Structure ◽  
Rna Structure ◽  
Structure Prediction ◽  
Rna Secondary Structure ◽  
Three Dimensional ◽  
Coarse Grained ◽  
Supplementary Information ◽  
Tree Graph ◽  
Rna Structure Prediction ◽  
Tree Graphs

Abstract Summary We launch a webserver for RNA structure prediction and design corresponding to tools developed using our RNA-As-Graphs (RAG) approach. RAG uses coarse-grained tree graphs to represent RNA secondary structure, allowing the application of graph theory to analyze and advance RNA structure discovery. Our webserver consists of three modules: (a) RAG Sampler: samples tree graph topologies from an RNA secondary structure to predict corresponding tertiary topologies, (b) RAG Builder: builds three-dimensional atomic models from candidate graphs generated by RAG Sampler, and (c) RAG Designer: designs sequences that fold onto novel RNA motifs (described by tree graph topologies). Results analyses are performed for further assessment/selection. The Results page provides links to download results and indicates possible errors encountered. RAG-Web offers a user-friendly interface to utilize our RAG software suite to predict and design RNA structures and sequences. Availability and implementation The webserver is freely available online at: http://www.biomath.nyu.edu/ragtop/. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Fractal behaviours of networks induced on infinite tree structures by random walks

2021 ◽  
Vol 2090 (1) ◽  
pp. 012085
Author(s):  
Nobutoshi Ikeda
Keyword(s):  
Random Walks ◽  
Small World ◽  
Numerical Results ◽  
Tree Structure ◽  
Box Dimension ◽  
Tree Graph ◽  
Tree Structures ◽  
Scale Free ◽  
Tree Graphs ◽  
Original Tree

Abstract Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

The harmonic index of unicyclic graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750039 ◽  
Author(s):  
R. Rasi ◽  
S. M. Sheikholeslami
Keyword(s):  
Maximum Degree ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Harmonic Index ◽  
Degree Of A Vertex

The harmonic index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. Hu and Zhou [WSEAS Trans. Math. 12 (2013) 716–726] proved that for any unicyclic graph [Formula: see text] of order [Formula: see text], [Formula: see text] with equality if and only if [Formula: see text]. Recently, Zhong and Cui [Filomat 29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph [Formula: see text] of order [Formula: see text] other than [Formula: see text], [Formula: see text]. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph [Formula: see text] of order [Formula: see text] with maximum degree [Formula: see text], [Formula: see text] and classify the extremal unicyclic graphs.

scholarly journals Algorithms for generalized stability numbers of tree graphs

1966 ◽  
Vol 6 (1) ◽  
pp. 89-100 ◽  
Author(s):  
D. E. Daykin ◽  
C. P. Ng
Keyword(s):  
Tree Graph ◽  
Directed Tree ◽  
External Stability ◽  
Tree Graphs ◽  
Internal And External Stability ◽  
Generalized Stability

In this paper we give some algorithms for determining αw(T) and βw(T), the generalized internal and external stability numbers respectively, of a finite directed tree graph T whose nodes are weighted by a function w. We define αw(T) and βw in section 2. When w gives every node of T the weight 1 then αw(T) = α(T) and βw(T) = β(T) where α(T) and β(T) are the usual stability numbers.

scholarly journals Lower Bounds on the Entire Zagreb Indices of Trees

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Liang Luo ◽  
Nasrin Dehgardi ◽  
Asfand Fahad
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Zagreb Indices ◽  
Degree Of A Vertex

For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑x is either adjacent or incident to ydxdy in which dx represents the degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.

scholarly journals A Note on Full and Complete Binary Planar Graphs

2020 ◽  
Vol 3 (2) ◽  
pp. 70
Author(s):  
Emily L Casinillo ◽  
Leomarich F Casinillo
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Binary Tree ◽  
Planar Graphs ◽  
Tree Graph ◽  
Complete Binary Tree ◽  
Combinatorial Formula ◽  
Acyclic Graph ◽  
Tree Graphs ◽  
Vertex Set

<p>Let G=(V(G), E(G)) be a connected graph where V(G) is a finite nonempty set called vertex-set of G, and  E(G) is a set of unordered pairs {u, v} of distinct elements from  V(G) called the edge-set of G. If  is a connected acyclic graph or a connected graph with no cycles, then it is called a tree graph. A binary tree Tl with l levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side. If we form a path on each level of a full and complete binary tree, then the graph is now called full and complete binary planar graph and it is denoted as Bn, where n is the level of the graph. This paper introduced a new planar graph which is derived from binary tree graphs. In addition, a combinatorial formula for counting its vertices, faces, and edges that depends on the level of the graph was developed.</p>

Synthesis of Kinematic Chains via Method of Cut-Set Matrix With Modulo-2 Operation

1973 ◽  
Vol 95 (3) ◽  
pp. 681-684 ◽  
Author(s):  
M. Huang ◽  
A. H. Soni
Keyword(s):  
Computer Aided Design ◽  
Tree Graph ◽  
Kinematic Chains ◽  
Tree Graphs ◽  
Computer Aided ◽  
Cut Set ◽  
Aided Design ◽  
High Degree

Method of cut-set matrix with modulo-2 operation is used to enumerate all possible tree graphs from a given tree graph. The typical kinematic chains corresponding to the tree graphs are the chains with cams and gears. The equations used to find the number of heavy edges (corresponding to cam or gear joints) and fine edges (corresponding to turning joints) required in the tree graphs are presented. A table of tree graphs and their corresponding cam kinematic chains is prepared. The method is potentially promising for computer-aided design application because of its simplicity, compact notation, and high degree of organization.

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