Cliques in Graphs With Bounded Minimum Degree

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

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Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽

10.3390/sym11121529 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1529 ◽

Cited By ~ 5

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal Ahmad Ganie ◽

Yilun Shang

Keyword(s):

Spectral Radius ◽

Undirected Graph ◽

Minimum Degree ◽

Distance Matrix ◽

Laplacian Matrix ◽

Extremal Graphs ◽

Special Cases ◽

Signless Laplacian ◽

Graph Parameters ◽

Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

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Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Symmetry ◽

10.3390/sym11050615 ◽

2019 ◽

Vol 11 (5) ◽

pp. 615

Author(s):

Hongzhuan Wang ◽

Piaoyang Yin

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Extremal Graphs ◽

Graph Invariant ◽

Resistance Distance ◽

Harary Index ◽

H Index ◽

Electronic Networks ◽

Minimum Number ◽

Graph Parameters

Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

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On the sum-connectivity index

Filomat ◽

10.2298/fil1103029w ◽

2011 ◽

Vol 25 (3) ◽

pp. 29-42 ◽

Cited By ~ 12

Author(s):

Shilin Wang ◽

Zhou Bo ◽

Nenad Trinajstic

Keyword(s):

Lower Bound ◽

Minimum Degree ◽

Simple Graph ◽

Connectivity Index ◽

Extremal Graphs ◽

Free Graph ◽

Mathematical Chemistry ◽

Degree Of Vertex

The sum-connectivity index of a simple graph G is defined in mathematical chemistry as R+(G) = ? uv?E(G)(du+dv)?1/2, where E(G) is the edge set of G and du is the degree of vertex u in G. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with n vertices and minimum degree at least two and characterize the extremal graphs, where n ? 11.

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THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE

Forum of Mathematics Pi ◽

10.1017/fmp.2020.7 ◽

2020 ◽

Vol 8 ◽

Author(s):

HONG LIU ◽

OLEG PIKHURKO ◽

KATHERINE STADEN

Keyword(s):

Exact Solution ◽

Edge Density ◽

Extremal Graphs ◽

Large Graphs ◽

Minimum Number

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$ , which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

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Triangles in Regular Graphs with Density Below One Half

Combinatorics Probability Computing ◽

10.1017/s0963548309009857 ◽

2009 ◽

Vol 18 (3) ◽

pp. 435-440 ◽

Cited By ~ 2

Author(s):

ALLAN SIU LUN LO

Keyword(s):

Regular Graphs ◽

Extremal Graphs ◽

Minimum Number

Let k3reg(n, d) be the minimum number of triangles in d-regular graphs with n vertices. We find the exact value of k3reg(n, d) for d between $2n /5 + 12 \sqrt{n}/5$ and n/2. In addition, we identify the structure of the extremal graphs.

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On Universal Threshold Graphs

Combinatorics Probability Computing ◽

10.1017/s096354830000122x ◽

1994 ◽

Vol 3 (3) ◽

pp. 327-344 ◽

Cited By ~ 7

Author(s):

P. L. Hammer ◽

A. K. Kelmans

Keyword(s):

Total Weight ◽

Stable Set ◽

Extremal Graphs ◽

Recursive Procedure ◽

Induced Subgraph ◽

Boolean Cube ◽

Threshold Graphs ◽

Minimum Number ◽

Universal Graphs ◽

Threshold Graph

A graph G is threshold if there exists a ‘weight’ function w: V(G) → R such that the total weight of any stable set of G is less than the total weight of any non-stable set of G. Let n denote the set of threshold graphs with n vertices. A graph is called n-universal if it contains every threshold graph with n vertices as an induced subgraph. n-universal threshold graphs are of special interest, since they are precisely those n-universal graphs that do not contain any non-threshold induced subgraph.In this paper we shall study minimumn-universal (threshold) graphs, i.e.n-universal (threshold) graphs having the minimum number of vertices.It is shown that for any n ≥ 3 there exist minimum n-universal graphs, which are themselves threshold, and others which are not.Two extremal minimum n-universal graphs having respectively the minimum and the maximum number of edges are described, it is proved that they are unique, and that they are threshold graphs.The set of all minimum n-universal threshold graphs is then described constructively; it is shown that it forms a lattice isomorphic to the n − 1 dimensional Boolean cube, and that the minimum and the maximum elements of this lattice are the two extremal graphs introduced above.The proofs provide a (polynomial) recursive procedure that determines for any threshold graph G with n vertices and for any minimum n-universal threshold graph T, an induced subgraph G' of T isomorphic to G.

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Bounds on the Arithmetic-Geometric Index

Symmetry ◽

10.3390/sym13040689 ◽

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.

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Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3198 ◽

2014 ◽

Vol 21 (1) ◽

Cited By ~ 1

Author(s):

Michael Krivelevich ◽

Wojciech Samotij

Keyword(s):

Random Graph ◽

Classical Result ◽

Minimum Degree ◽

Average Degree ◽

Free Graph ◽

Connected Graphs ◽

Random Subgraph ◽

Minimum Number ◽

Random Subgraphs ◽

Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

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On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2149 ◽

2016 ◽

Vol Vol. 17 no. 3 (Graph Theory) ◽

Author(s):

Shih-Yan Chen ◽

Shin-Shin Kao ◽

Hsun Su

Keyword(s):

Minimum Degree ◽

Degree Sequence ◽

Sequence Characterization ◽

Hamiltonian Connected ◽

Vertex Graph ◽

Minimum Number ◽

Hamiltonian Connectivity ◽

International Audience ◽

Delta G ◽

Hamiltonian Properties

International audience Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, or $k$-vertex fault Hamiltonian-connected if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

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