scholarly journals Extremal graphs with respect to vertex betweenness centrality for certain graph families

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3123-3130
Author(s):  
Jelena Klisara ◽  
Jana Hurajová ◽  
Tomás Madaras ◽  
Riste Skrekovski
Keyword(s):  
Betweenness Centrality ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Shortest Paths ◽  
Extremal Graphs ◽  
Wheel Graph ◽  
Graph Families ◽  
Pass Through ◽  
Fan Graph ◽  
Extremal Values

The betweenness centrality of a vertex in a graph is the sum of relative numbers of shortest paths that pass through that vertex. We study extremal values of vertex betweenness within various families of graphs. We prove that, in the family of 2-connected (resp. 3-connected) graphs on n vertices, the maximum betweenness value is reached for the maximum degree vertex of the fan graph F1,n-1 (resp. the wheel graph Wn); the maximum betweenness values, their realizing vertices and extremal graphs are determined also for wider families of graphs of minimum degree at least 2 or 3, respectively, and, in addition, for graphs with prescribed maximum degree or prescribed diameter at least 3.

scholarly journals Bounds on the Arithmetic-Geometric Index

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 689
Author(s):  
José M. Rodríguez ◽  
José L. Sánchez ◽  
José M. Sigarreta ◽  
Eva Tourís
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Point Of View ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Star Graphs ◽  
Practical Point ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Atom Bond

The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize the extremal graphs with respect to them. Several bounds are based on other indices, such as the second variable Zagreb index or the general atom-bond connectivity index), and some of them involve some parameters, such as the number of edges, the maximum degree, or the minimum degree of the graph. In most bounds, the graphs for which equality is attained are regular or biregular, or star graphs.

scholarly journals Betweenness centrality for temporal multiplexes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Silvia Zaoli ◽  
Piero Mazzarisi ◽  
Fabrizio Lillo
Keyword(s):  
Information Flow ◽  
Real World ◽  
Betweenness Centrality ◽  
Temporal Structure ◽  
Shortest Paths ◽  
Single Layer ◽  
Distance Metric ◽  
Definition Of

AbstractBetweenness centrality quantifies the importance of a vertex for the information flow in a network. The standard betweenness centrality applies to static single-layer networks, but many real world networks are both dynamic and made of several layers. We propose a definition of betweenness centrality for temporal multiplexes. This definition accounts for the topological and temporal structure and for the duration of paths in the determination of the shortest paths. We propose an algorithm to compute the new metric using a mapping to a static graph. We apply the metric to a dataset of $$\sim 20$$ ∼ 20 k European flights and compare the results with those obtained with static or single-layer metrics. The differences in the airports rankings highlight the importance of considering the temporal multiplex structure and an appropriate distance metric.

scholarly journals Cliques in Graphs With Bounded Minimum Degree

2012 ◽  
Vol 21 (3) ◽  
pp. 457-482 ◽  
Author(s):  
ALLAN LO
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Minimum Number

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).

scholarly journals The Power of Quasi-Shortest Paths: $\rho$ -Geodesic Betweenness Centrality

2017 ◽  
Vol 4 (3) ◽  
pp. 187-200
Author(s):  
Dianne S. V. de Medeiros ◽  
Miguel Elias M. Campista ◽  
Nathalie Mitton ◽  
Marcelo Dias de Amorim ◽  
Guy Pujolle
Keyword(s):  
Betweenness Centrality ◽  
Shortest Paths

scholarly journals Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11

2014 ◽  
Vol 315-316 ◽  
pp. 128-134 ◽  
Author(s):  
O.V. Borodin ◽  
A.O. Ivanova ◽  
A.V. Kostochka
Keyword(s):  
Maximum Degree ◽  
Minimum Degree

Semi-Deterministic Construction of Scale-Free Networks with Designated Parameters

2018 ◽  
Vol 18 (01) ◽  
pp. 1850001
Author(s):  
NAOKI TAKEUCHI ◽  
SATOSHI FUJITA
Keyword(s):  
Degree Distribution ◽  
Interconnection Networks ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Scale Free ◽  
Scale Free Networks ◽  
Ideal Number ◽  
Message Propagation ◽  
Deterministic Construction ◽  
The Ideal

Scale-free networks have several favorable properties as the topology of interconnection networks such as the short diameter and the quick message propagation. In this paper, we propose a method to construct scale-free networks in a semi-deterministic manner. The proposed algorithm extends the Bulut's algorithm for constructing scale-free networks with designated minimum degree k and maximum degree m, in such a way that: (1) it determines the ideal number of edges derived from the ideal degree distribution; and (2) after connecting each new node to k existing nodes as in the Bulut’s algorithm, it adjusts the number of edges to the ideal value by conducting add/removal of edges. We prove that such an adjustment is always possible if the number of nodes in the network exceeds [Formula: see text]. The performance of the algorithm is experimentally evaluated.

scholarly journals Identifying Node Importance in a Complex Network Based on Node Bridging Feature

2018 ◽  
Vol 8 (10) ◽  
pp. 1914 ◽  
Author(s):  
Lincheng Jiang ◽  
Yumei Jing ◽  
Shengze Hu ◽  
Bin Ge ◽  
Weidong Xiao
Keyword(s):  
Complex Networks ◽  
Large Scale ◽  
Shortest Paths ◽  
Evaluation Criteria ◽  
Local Network ◽  
Network Efficiency ◽  
Network Attack ◽  
Node Importance ◽  
Pass Through ◽  
Large Scale Networks

Identifying node importance in complex networks is of great significance to improve the network damage resistance and robustness. In the era of big data, the size of the network is huge and the network structure tends to change dynamically over time. Due to the high complexity, the algorithm based on the global information of the network is not suitable for the analysis of large-scale networks. Taking into account the bridging feature of nodes in the local network, this paper proposes a simple and efficient ranking algorithm to identify node importance in complex networks. In the algorithm, if there are more numbers of node pairs whose shortest paths pass through the target node and there are less numbers of shortest paths in its neighborhood, the bridging function of the node between its neighborhood nodes is more obvious, and its ranking score is also higher. The algorithm takes only local information of the target nodes, thereby greatly improving the efficiency of the algorithm. Experiments performed on real and synthetic networks show that the proposed algorithm is more effective than benchmark algorithms on the evaluation criteria of the maximum connectivity coefficient and the decline rate of network efficiency, no matter in the static or dynamic attack manner. Especially in the initial stage of attack, the advantage is more obvious, which makes the proposed algorithm applicable in the background of limited network attack cost.

scholarly journals Betweenness centrality profiles in trees

2017 ◽  
Vol 5 (5) ◽  
pp. 776-794
Author(s):  
Benjamin Fish ◽  
Rahul Kushwaha ◽  
György Turán
Keyword(s):  
Betweenness Centrality ◽  
Shortest Paths ◽  
Random Trees ◽  
Experimental Results ◽  
Basic Notion ◽  
Worst Case ◽  
Scale Free ◽  
Order Of Magnitude

Abstract Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The $k$-betweenness centrality of a vertex is defined similarly, but only considers shortest paths of length at most $k$. The sequence of $k$-betweenness centralities for all possible values of $k$ forms the betweenness centrality profile of a vertex. We study properties of betweenness centrality profiles in trees. We show that for scale-free random trees, for fixed $k$, the expectation of $k$-betweenness centrality strictly decreases as the index of the vertex increases. We also analyse worst-case properties of profiles in terms of the distance of profiles from being monotone, and the number of times pairs of profiles can cross. This is related to whether $k$-betweenness centrality, for small values of $k$, may be used instead of having to consider all shortest paths. Bounds are given that are optimal in order of magnitude. We also present some experimental results for scale-free random trees.

scholarly journals On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

10.37236/5173 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Special Class ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Approach ◽  
The Family ◽  
Edge Colourings ◽  
Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

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