scholarly journals On Compact Symmetric Regularizations of Graphs

10.37236/3821 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
R. Vandell ◽  
M. Walsh ◽  
W. D. Weakley
Keyword(s):  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Cartesian Product ◽  
Simple Graph ◽  
Induced Subgraph ◽  
Product Construction

Let $G$ be a finite simple graph of order $n$, maximum degree $\Delta$, and minimum degree $\delta$. A compact regularization of $G$ is a $\Delta$-regular graph $H$ of which $G$ is an induced subgraph: $H$ is symmetric if every automorphism of $G$ can be extended to an automorphism of $H$. The index $|H:G|$ of a regularization $H$ of $G$ is the ratio $|V(H)|/|V(G)|$. Let $\mbox{mcr}(G)$ denote the index of a minimum compact regularization of $G$ and let $\mbox{mcsr}(G)$ denote the index of a minimum compact symmetric regularization of $G$.Erdős and Kelly proved that every graph $G$ has a compact regularization and $\mbox{mcr}(G) \leq 2$. Building on a result of König, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and $\mbox{mcsr}(G) \leq 2^{\Delta - \delta}$.  Using a partial Cartesian product construction, we improve this to $\mbox{mcsr}(G) \leq \Delta - \delta + 2$ and give examples to show this bound cannot be reduced below $\Delta - \delta + 1$.

Component factors of the Cartesian product of graphs

2016 ◽  
Vol 09 (02) ◽  
pp. 1650041
Author(s):  
M. R. Chithra ◽  
A. Vijayakumar
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Induced Subgraph ◽  
Connected Graphs ◽  
Cartesian Product Of Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Product Of Graphs

Let [Formula: see text] be a family of connected graphs. A spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor (component factor) of [Formula: see text] if each component of [Formula: see text] is in [Formula: see text]. In this paper, we study the component factors of the Cartesian product of graphs. Here, we take [Formula: see text] and show that every connected graph [Formula: see text] has a [Formula: see text]-factor where [Formula: see text] and [Formula: see text] is the maximum degree of an induced subgraph [Formula: see text] in [Formula: see text] or [Formula: see text]. Also, we characterize graphs [Formula: see text] having a [Formula: see text]-factor.

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>

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

1980 ◽  
Vol 32 (6) ◽  
pp. 1325-1332 ◽  
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,

scholarly journals On inverse degree and topological indices of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2111-2120 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Jinlan Wang
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Topological Indices ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
Inverse Degree ◽  
Abc Index

Let G=(V,E) be a simple graph of order n and size m with maximum degree ? and minimum degree ?. The inverse degree of a graph G with no isolated vertices is defined as ID(G) = ?n,i=1 1/di, where di is the degree of the vertex vi?V(G). In this paper, we obtain several lower and upper bounds on ID(G) of graph G and characterize graphs for which these bounds are best possible. Moreover, we compare inverse degree ID(G) with topological indices (GA1-index, ABC-index, Kf-index) of graphs.

scholarly journals Perfect Matchings in $\epsilon$-regular Graphs

10.37236/1351 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Noga Alon ◽  
Vojtech Rödl ◽  
Andrzej Ruciński
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

scholarly journals Colouring the Petals of a Graph

10.37236/1699 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Cariolaro ◽  
Gianfranco Cariolaro
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Petersen Graph ◽  
Empty Graph ◽  
Single Exception ◽  
Class 1

A petal graph is a connected graph $G$ with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a $2$–regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class $1$. This settles a particular case of a conjecture of Hilton and Zhao.

scholarly journals New Graph Of Graph

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Domination Number ◽  
Clique Number ◽  
External Relation ◽  
Internal Relation ◽  
Coloring Number ◽  
As Graph ◽  
The Way

Constructing new graph from the graph's parameters and related notions in the way that, the study on the new graph and old graph in their parameters could be facilitated. As graph, new graph has some characteristics and results which are related to the structure of this graph. For this purpose, regular graph is considered so the internal relation and external relation on this new graph are studied. The kind of having same number of edges when this number is originated by common number of graphs like maximum degree, minimum degree, domination number, coloring number and clique number, is founded in the word of having regular graph

The f-Chromatic Index of a Graph Whose f-Core Has Maximum Degree 2

2013 ◽  
Vol 56 (3) ◽  
pp. 449-458 ◽  
Author(s):  
S. Akbari ◽  
M. Chavooshi ◽  
M. Ghanbari ◽  
S. Zare
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Disjoint Union ◽  
Minimum Degree ◽  
Chromatic Index ◽  
Induced Subgraph ◽  
Class 1 ◽  
Minimum Number

Abstract.Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by χ'(G). It is well known that , for any graph G, where Δ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x'(G) = Δ(G) and class 2 if χ'(G) = Δ(G)+1. Also, GΔ is the induced subgraph on all vertices of degree Δ(G). Let f : V(G) → ℕ be a function. An f-coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v ∊ V(G) at most f (v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by χ'f (G). It was shown that for every graph , where . A graph G is said to be f -class 1 , and f -class 2, otherwise. Also, GΔf is the induced subgraph of G on . Hilton and Zhao showed that if G has maximum degree two and G is class 2, then G is critical, GΔ is a disjoint union of cycles and δ(G) = Δ(G)–1, where δ(G) denotes the minimum degree of G, respectively. In this paper, we generalize this theorem to f -coloring of graphs. Also, we determine the f -chromatic index of a connected graph G with |GΔf| ≤ 4.

scholarly journals Bounds for Incidence Energy of Some Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Weizhong Wang ◽  
Dong Yang
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Line Graph ◽  
Singular Values ◽  
Simple Graph

LetGbe a simple graph. The incidence energy (IEfor short) ofGis defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound forIEof graphs in terms of the maximum degree is given. Meanwhile, bounds forIEof the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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