scholarly journals On the adjacent eccentric distance sum of graphs

2014 ◽  
Vol 68 (2) ◽  
pp. 1-10
Author(s):  
Halina Bielak ◽  
Katarzyna Wolska
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Wiener Index ◽  
International Mathematical ◽  
Eccentric Distance

AbstractIn this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum. Vol. 7 (2O02) no. 26. 1280-1294].The adjaceni eccentric distance sum index of the graph G is defined aswhere ε(υ) is the eccentricity of the vertex υ, deg(υ) is the degree of the vertex υ and D(υ) = ∑

scholarly journals Wiener index in graphs with given minimum degree and maximum degree

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Dankelmann ◽  
Alex Alochukwu
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Large Degree ◽  
Wiener Index ◽  
Sharp Bound

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.

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 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].

Extremal trees with respect to the Steiner Wiener index

2019 ◽  
Vol 11 (06) ◽  
pp. 1950067
Author(s):  
Jie Zhang ◽  
Guang-Jun Zhang ◽  
Hua Wang ◽  
Xiao-Dong Zhang
Keyword(s):  
Maximum Degree ◽  
Computational Analysis ◽  
Degree Sequence ◽  
Wiener Index ◽  
Extremal Problems ◽  
Difficult Problem ◽  
Computational Results ◽  
Number Of Leaves ◽  
Extremal Structure ◽  
Chemical Trees

The well-known Wiener index is defined as the sum of pairwise distances between vertices. Extremal problems with respect to it have been extensively studied for trees. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of minimum sizes of subgraphs that span [Formula: see text] given vertices over all possible choices of the [Formula: see text] vertices. We consider the extremal problems with respect to the Steiner Wiener index among trees of a given degree sequence. First, it is pointed out minimizing the Steiner Wiener index in general may be a difficult problem, although the extremal structure may very likely be the same as that for the regular Wiener index. We then consider the upper bound of the general Steiner Wiener index among trees of a given degree sequence and study the corresponding extremal trees. With these findings, some further discussion and computational analysis are presented for chemical trees. We also propose a conjecture based on the computational results. In addition, we identify the extremal trees that maximize the Steiner Wiener index among trees with a given maximum degree or number of leaves.

Pitchfork domination in graphs

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?

scholarly journals A spanning bandwidth theorem in random graphs

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.

scholarly journals Bounds of modified Sombor index, spectral radius and energy

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>

scholarly journals Bounds for symmetric division deg index of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 683-698 ◽  
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.

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