Computing the Edge-Neighbour-Scattering Number of Graphs

2013 ◽  
Vol 68 (10-11) ◽  
pp. 599-604
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Single Vertex ◽  
Number Of Components ◽  
Np Complete

A set of edges X is subverted from a graph G by removing the closed neighbourhood N[X] from G. We denote the survival subgraph by G=X. An edge-subversion strategy X is called an edge-cut strategy of G if G=X is disconnected, a single vertex, or empty. The edge-neighbour-scattering number of a graph G is defined as ENS(G) = max{ω(G/X)-|X| : X is an edge-cut strategy of G}, where w(G=X) is the number of components of G=X. This parameter can be used to measure the vulnerability of networks when some edges are failed, especially spy networks and virus-infected networks. In this paper, we prove that the problem of computing the edge-neighbour-scattering number of a graph is NP-complete and give some upper and lower bounds for this parameter.

Vertex-Neighbor-Scattering Number of Bipartite Graphs

2016 ◽  
Vol 27 (04) ◽  
pp. 501-509
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Number Of Components ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

Let G be a connected graph. A set of vertices [Formula: see text] is called subverted from G if each of the vertices in S and the neighbor of S in G are deleted from G. By G/S we denote the survival subgraph that remains after S is subverted from G. A vertex set S is called a cut-strategy of G if G/S is disconnected, a clique, or ø. The vertex-neighbor-scattering number of G is defined by [Formula: see text], where S is any cut-strategy of G, and ø(G/S) is the number of components of G/S. It is known that this parameter can be used to measure the vulnerability of spy networks and the computing problem of the parameter is NP-complete. In this paper, we discuss the vertex-neighbor-scattering number of bipartite graphs. The NP-completeness of the computing problem of this parameter is proven, and some upper and lower bounds of the parameter are also given.

scholarly journals Metric dimension of Andrásfai graphs

2019 ◽  
Vol 39 (3) ◽  
pp. 415-423
Author(s):  
S. Batool Pejman ◽  
Shiroyeh Payrovi ◽  
Ali Behtoei
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Metric Dimension ◽  
Resolving Set ◽  
Np Complete ◽  
General Graphs

A set \(W\subseteq V(G)\) is called a resolving set, if for each pair of distinct vertices \(u,v\in V(G)\) there exists \(t\in W\) such that \(d(u,t)\neq d(v,t)\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The cardinality of a minimum resolving set for \(G\) is called the metric dimension of \(G\) and is denoted by \(\dim_M(G)\). This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and \(And(k)\square P_n\). Also, we provide upper and lower bounds for \(dim_M(And(k)\square C_n)\).

scholarly journals Injective edge coloring of graphs

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6411-6423
Author(s):  
Domingos Cardoso ◽  
Orestes Cerdeira ◽  
Charles Dominicc ◽  
Pedro Cruz
Keyword(s):  
Lower Bounds ◽  
Edge Coloring ◽  
Upper And Lower Bounds ◽  
Coloring Number ◽  
Minimum Number ◽  
Np Complete

Three edges e1, e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then c(e1)? c(e3). The injective edge coloring number ?'i(G) is the minimum number of colors permitted in such a coloring. In this paper, exact values of ?'i(G) for several classes of graphs are obtained, upper and lower bounds for ?'i(G) are introduced and it is proven that checking whether ?'i(G) = k is NP-complete.

scholarly journals From Edge-Coloring to Strong Edge-Coloring

10.37236/3529 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Valentin Borozan ◽  
Gerard Jennhwa Chang ◽  
Nathann Cohen ◽  
Shinya Fujita ◽  
Narayanan Narayanan ◽  
...  
Keyword(s):  
Set Theory ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Upper And Lower Bounds ◽  
Combinatorial Set ◽  
Proper Edge Coloring ◽  
Complete Bipartite Graphs ◽  
Np Complete ◽  
Subcubic Graphs ◽  
Combinatorial Set Theory

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: $k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.An addendum was added to this paper on Jul 4, 2015.

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.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
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.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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