On vertex partitions and some minor-monotone graph parameters

AbstractThere has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

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On the Relation Between Two Minor-Monotone Graph Parameters

COMBINATORICA ◽

10.1007/pl00009821 ◽

1998 ◽

Vol 18 (2) ◽

pp. 281-292 ◽

Cited By ~ 3

Author(s):

R. Pendavingh

Keyword(s):

Graph Parameters ◽

Monotone Graph

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On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽

10.3390/math9050512 ◽

2021 ◽

Vol 9 (5) ◽

pp. 512

Author(s):

Maryam Baghipur ◽

Modjtaba Ghorbani ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Distance Matrix ◽

Upper And Lower Bounds ◽

Laplacian Eigenvalue ◽

Signless Laplacian ◽

Graph Parameters ◽

Simple Connected Graph ◽

Second Largest Eigenvalue ◽

Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

Symmetry ◽

10.3390/sym13061036 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1036

Author(s):

Abel Cabrera Martínez ◽

Alejandro Estrada-Moreno ◽

Juan Alberto Rodríguez-Velázquez

Keyword(s):

Vertex Cover ◽

Domination Number ◽

Theoretical Computer Science ◽

Discrete Mathematics ◽

Special Issue ◽

Total Domination Number ◽

Theoretical Computer ◽

Graph Parameters ◽

Semitotal Domination ◽

Domination In Graphs

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.

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Vertex partitions of r-edge-colored graphs

Applied Mathematics-A Journal of Chinese Universities ◽

10.1007/s11766-008-0117-0 ◽

2008 ◽

Vol 23 (1) ◽

pp. 120-126

Author(s):

Ze-min Jin ◽

Xue-liang Li

Keyword(s):

Colored Graphs ◽

Vertex Partitions

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Cubical coloring — fractional covering by cuts and semidefinite programming

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2134 ◽

2015 ◽

Vol Vol. 17 no.2 (Graph Theory) ◽

Author(s):

Robert Šámal

Keyword(s):

Semidefinite Programming ◽

Chromatic Number ◽

Fractional Chromatic Number ◽

Maximum Cut ◽

Continuous Mappings ◽

Polynomial Time Approximation Algorithm ◽

Eigenvalue Bound ◽

International Audience ◽

Graph Parameters ◽

Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

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The Impact of Edge Correlations in Random Networks

Complex Systems ◽

10.25088/complexsystems.30.4.525 ◽

2021 ◽

Vol 30 (4) ◽

pp. 525-537

Author(s):

András Faragó ◽

Keyword(s):

Random Graph ◽

Random Graphs ◽

Real Life ◽

Graph Model ◽

Random Networks ◽

Random Graph Model ◽

Special Cases ◽

Graph Parameters ◽

The Impact ◽

Good Number

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

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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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Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

Special Matrices ◽

10.1515/spma-2018-0028 ◽

2018 ◽

Vol 6 (1) ◽

pp. 343-356

Author(s):

K. Arathi Bhat ◽

G. Sudhakara

Keyword(s):

Chromatic Number ◽

Perfect Matching ◽

Domination Number ◽

Perfect Matchings ◽

Factor Graph ◽

Vertex Set ◽

Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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On the relations of graph parameters and its total parameters

Acta Mathematicae Applicatae Sinica English Series ◽

10.1007/s10255-008-8152-4 ◽

2009 ◽

Vol 26 (3) ◽

pp. 525-528

Author(s):

Jing-wen Li ◽

Zhi-wen Wang ◽

Jaeun Lee ◽

Moo Young Sohn ◽

Zhong-fu Zhang ◽

...

Keyword(s):

Graph Parameters

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