Average eccentricity, minimum degree and maximum degree in graphs

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.

Download Full-text

On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

The Electronic Journal of Combinatorics ◽

10.37236/5173 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 8

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].

Download Full-text

Pitchfork domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500251 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050025

Author(s):

Manal N. Al-Harere ◽

Mohammed A. Abdlhusein

Keyword(s):

Maximum Degree ◽

Undirected Graph ◽

Minimum Degree ◽

Dominating Set ◽

Domination Number ◽

Dominating Sets ◽

Minimum Cardinality ◽

New Model ◽

Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

Download Full-text

A spanning bandwidth theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548321000481 ◽

2021 ◽

pp. 1-31

Author(s):

Peter Allen ◽

Julia Böttcher ◽

Julia Ehrenmüller ◽

Jakob Schnitzer ◽

Anusch Taraz

Keyword(s):

Random Graph ◽

Random Graphs ◽

Maximum Degree ◽

Minimum Degree ◽

Spanning Subgraph ◽

Vertex Graph ◽

Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

Download Full-text

Bounds of modified Sombor index, spectral radius and energy

AIMS Mathematics ◽

10.3934/math.2021653 ◽

2021 ◽

Vol 6 (10) ◽

pp. 11263-11274

Author(s):

Yufei Huang ◽

◽

Hechao Liu ◽

Keyword(s):

Spectral Radius ◽

Maximum Degree ◽

Minimum Degree ◽

Simple Graph ◽

Degree Of Vertex ◽

The Relationship

<abstract><p>Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.</p></abstract>

Download Full-text

Bounds for symmetric division deg index of graphs

Filomat ◽

10.2298/fil1903683d ◽

2019 ◽

Vol 33 (3) ◽

pp. 683-698 ◽

Cited By ~ 1

Author(s):

Kinkar Das ◽

Marjan Matejic ◽

Emina Milovanovic ◽

Igor Milovanovic

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Topological Index ◽

Minimum Degree ◽

Topological Indices ◽

Symmetric Division ◽

Mathematical Properties ◽

Vertex Degrees ◽

Additive Modeling ◽

Simple Connected Graph

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

Download Full-text

Line Graphs of Monogenic Semigroup Graphs

Journal of Mathematics ◽

10.1155/2021/6630011 ◽

2021 ◽

Vol 2021 ◽

pp. 1-4

Author(s):

Nihat Akgunes ◽

Yasar Nacaroglu ◽

Sedat Pak

Keyword(s):

Maximum Degree ◽

Zero Divisor ◽

Minimum Degree ◽

Line Graph ◽

Domination Number ◽

Clique Number ◽

Line Graphs ◽

Monogenic Semigroup ◽

Graph Properties ◽

Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

Download Full-text

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-102-7 ◽

1980 ◽

Vol 32 (6) ◽

pp. 1325-1332 ◽

Cited By ~ 5

Author(s):

J. A. Bondy ◽

R. C. Entringer

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Minimum Degree ◽

Simple Graph ◽

General Context ◽

Connected Graphs ◽

Longest Cycles ◽

Image Position ◽

The Relationship

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

Download Full-text

Inverse Sum Indeg Index of Subdivision, t-Subdivision Graphs, and Related Sums

Handbook of Research on Advanced Applications of Graph Theory in Modern Society - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-5225-9380-5.ch005 ◽

2020 ◽

pp. 104-119 ◽

Cited By ~ 1

Author(s):

Amitav Doley ◽

Jibonjyoti Buragohain ◽

A. Bharali

Keyword(s):

Surface Area ◽

Maximum Degree ◽

Minimum Degree ◽

Total Surface Area ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Zagreb Index ◽

Octane Isomers ◽

Graph Parameters ◽

The First Zagreb Index

The inverse sum indeg (ISI) index of a graph G is defined as the sum of the weights dG(u)dG(v)/dG(u)+dG(v) of all edges uv in G, where dG(u) is the degree of the vertex u in G. This index is found to be a significant predictor of total surface area of octane isomers. In this chapter, the authors present some lower and upper bounds for ISI index of subdivision graphs, t-subdivision graphs, s-sum and st -sum of graphs in terms of some graph parameters such as order, size, maximum degree, minimum degree, and the first Zagreb index. The extremal graphs are also characterized for their sharpness.

Download Full-text