scholarly journals General Properties on Differential Sets of a Graph

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 265
Author(s):  
Ludwin A. Basilio ◽  
Sergio Bermudo ◽  
Juan C. Hernández-Gómez ◽  
José M. Sigarreta
Keyword(s):  
Lower Bounds ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
Limited Resources ◽  
Maximization Problem ◽  
General Properties ◽  
Coverage Maximization ◽  
Service Coverage

Let G=(V,E) be a graph, and let β∈R. Motivated by a service coverage maximization problem with limited resources, we study the β-differential of G. The β-differential of G, denoted by ∂β(G), is defined as ∂β(G):=max{|B(S)|−β|S|suchthatS⊆V}. The case in which β=1 is known as the differential of G, and hence ∂β(G) can be considered as a generalization of the differential ∂(G) of G. In this paper, upper and lower bounds for ∂β(G) are given in terms of its order |G|, minimum degree δ(G), maximum degree Δ(G), among other invariants of G. Likewise, the β-differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have.

scholarly journals The treewidth of 2-section of hypergraphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Ke Liu ◽  
Mei Lu
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
Average Rank ◽  
Linear Hypergraph ◽  
Algorithmic Graph Theory

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

scholarly journals Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1711
Author(s):  
Zhao Wang ◽  
Yaping Mao ◽  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
K Index ◽  
Gutman Index

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=∑S⊆V(G),|S|=k∏v∈SdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.

scholarly journals Remarks on upper and lower bounds formatching sequencibility of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2091-2099
Author(s):  
Shuya Chiba ◽  
Yuji Nakano
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Hamilton Cycle ◽  
Upper And Lower Bounds ◽  
Linear Ordering ◽  
The Relationship

In 2008, Alspach [The Wonderful Walecki Construction, Bull. Inst. Combin. Appl. 52 (2008) 7-20] defined the matching sequencibility of a graph G to be the largest integer k such that there exists a linear ordering of its edges so that every k consecutive edges in the linear ordering form a matching of G, which is denoted by ms(G). In this paper, we show that every graph G of size q and maximum degree ? satisfies 1/2?q/?+1? ? ms(G) ? ?q?1/??1? by using the edge-coloring of G, and we also improve this lower bound for some particular graphs. We further discuss the relationship between the matching sequencibility and a conjecture of Seymour about the existence of the kth power of a Hamilton cycle.

scholarly journals New Bounds for Topological Indices on Trees through Generalized Methods

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1097 ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Chemical Compounds ◽  
Wiener Index ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Unified Approach ◽  
Wide Family ◽  
New Inequalities

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.

scholarly journals Colouring the Square of the Cartesian Product of Trees

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
International Audience

Graphs and Algorithms International audience We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.

scholarly journals Dominating the Direct Product of Two Graphs through Total Roman Strategies

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1438 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael G. Yero
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Minimum Degree ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Product Graphs ◽  
Roman Dominating Function ◽  
Dominating Function

Given a graph G without isolated vertices, a total Roman dominating function for G is a function f:V(G)→{0,1,2} such that every vertex u with f(u)=0 is adjacent to a vertex v with f(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number γtR(G) of G is the smallest possible value of ∑v∈V(G)f(v) among all total Roman dominating functions f. The total Roman domination number of the direct product G×H of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between γtR(G×H) and some classical domination parameters for the factors are given. Characterizations of the direct product graphs G×H achieving small values (≤7) for γtR(G×H) are presented, and exact values for γtR(G×H) are deduced, while considering various specific direct product classes.

scholarly journals A short note on hyper Zagreb index

2017 ◽  
Vol 37 (2) ◽  
pp. 51-58
Author(s):  
Suresh Elumalai ◽  
Toufik Mansour ◽  
Mohammad Ali Rostami ◽  
Gnanadhass Britto Antony Xavier
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Short Note ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Counter Example ◽  
Inverse Degree

In this paper, we present and analyze the upper and lower bounds on the Hyper Zagreb index $\chi^2(G)$ of graph $G$ in terms of the number of vertices $(n)$, number of edges $(m)$, maximum degree $(\Delta)$, minimum degree $(\delta)$ and the inverse degree $(ID(G))$. In addition, we give a counter example on the upper bound  of the second Zagreb index for Theorems 2.2 and  2.4 from \cite{ranjini}. Finally, we present lower and upper bounds on $\chi^2(G)+\chi^2(\overline G)$, where $\overline G$ denotes the complement of $G$.

scholarly journals Anagram-Free Graph Colouring

10.37236/6267 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tim E. Wilson ◽  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Empty Word ◽  
Upper And Lower Bounds ◽  
Graph Colouring ◽  
Free Graph ◽  
Random Structures ◽  
Edge Colouring

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al.[Random Structures & Algorithms 2002] asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer  this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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