Neighbor Rupture Degree of Harary Graphs

2016 ◽  
Vol 27 (06) ◽  
pp. 739-756
Author(s):  
Ferhan Nihan Altundag ◽  
Goksen Bacak-Turan
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Maximum Order ◽  
Communication Links

The vulnerability shows the endurance of the network until the communication collapse after the breakdown of certain stations or communication links. If a spy or a station is invaded in a spy network, then the adjacent stations are treacherous. A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph G is defined to be [Formula: see text] where S is any vertex subversion strategy of G, w(G/S) is the number of connected components in G/S, and c(G/S) is the maximum order of the components of G/S. In this paper, the neighbor rupture degree of Harary graphs are obtained.

scholarly journals Graph Operations and Neighbor Rupture Degree

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Saadet Kandİlcİ ◽  
Goksen Bacak-Turan ◽  
Refet Polat
Keyword(s):  
Communication Network ◽  
Connected Components ◽  
Maximum Order ◽  
Graph Operations ◽  
Communication Links ◽  
Closed Neighborhood

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graph , say , is a set of vertices in whose closed neighborhood is removed from . The survival subgraph is denoted by . The neighbor rupture degree of , , is defined to be , where is any vertex subversion strategy of , is the number of connected components in and is the maximum order of the components of (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.

Neighbor Rupture Degree of Transformation Graphs Gxy−

2017 ◽  
Vol 28 (04) ◽  
pp. 335-355
Author(s):  
Goksen Bacak-Turan ◽  
Ekrem Oz
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Connected Components ◽  
Complete Graphs ◽  
Maximum Order ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs

A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] is any vertex subversion strategy of [Formula: see text], [Formula: see text] is the number of connected components in [Formula: see text], and [Formula: see text] is the maximum order of the components of [Formula: see text]. In this study, the neighbor rupture degree of transformation graphs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of path graphs, cycle graphs, wheel graphs, complete graphs and complete bipartite graphs are obtained.

The Maximum Genus of Cartesian Products of Graphs

1974 ◽  
Vol 26 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Open Disk ◽  
Maximum Genus ◽  
Cartesian Products ◽  
Euler’S Formula ◽  
Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

scholarly journals On the maximum orders of an induced forest, an induced tree, and a stable set

2014 ◽  
Vol 24 (2) ◽  
pp. 199-215
Author(s):  
Alain Hertz ◽  
Odile Marcotte ◽  
David Schindl
Keyword(s):  
Connected Graph ◽  
Stable Set ◽  
Stability Number ◽  
Maximum Order ◽  
The Stability ◽  
Special Case

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f - t is at most n - ?2?n-1?. In the special case where n is of the form a2 + 1 for some even integer a ? 4, f - t is at most n - ?2?n-1?-1. We also prove that these bounds are tight. In addition, letting ? denote the stability number of G, we show that ? - t is at most n + 1- ?2?2n? this bound is also tight.

Upper triangle free detour number of a graph

2021 ◽  
pp. 2150094
Author(s):  
S. Sethu Ramalingam ◽  
S. Athisayanathan
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Proper Subset ◽  
Maximum Order ◽  
Minimum Order ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers ◽  
Distance Formula

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

scholarly journals Undirecting membership in models of Anti-Foundation

2020 ◽  
Author(s):  
Bea Adam-Day ◽  
Peter J. Cameron
Keyword(s):  
Set Theory ◽  
Random Graph ◽  
Connected Graph ◽  
Countable Model ◽  
Connected Components ◽  
Membership Relation ◽  
Multiple Edges

AbstractIt is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

scholarly journals ALGEBRAIC CHARACTERIZATIONS OF GRAPH IMBEDDABILITY IN SURFACES AND PSEUDOSURFACES

2006 ◽  
Vol 15 (06) ◽  
pp. 681-693 ◽  
Author(s):  
LOWELL ABRAMS ◽  
DANIEL C. SLILATY
Keyword(s):  
Connected Graph ◽  
Homology Group ◽  
Dual Graph ◽  
Connected Components ◽  
Homology Groups ◽  
The Given

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first ℤ2 homology group. The characterizations are formulated in terms of the existence of a dual graph G* on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.

NEIGHBOR INTEGRITY OF TRANSFORMATION GRAPHS

2013 ◽  
Vol 24 (03) ◽  
pp. 303-317 ◽  
Author(s):  
GOKSEN BACAK-TURAN ◽  
ALPAY KIRLANGIC
Keyword(s):  
Communication Network ◽  
Maximum Order ◽  
Communication Links ◽  
The Stability

In a communication network, the vulnerability measures are essential to guide the designer in choosing an appropriate topology. They measure the stability of the network to disruption of operation after the failure of certain stations or communication links. If a station or operative is captured in a spy network, then the adjacent stations will be betrayed and are therefore useless in the whole network. In this sense, Margaret B. Cozzens and Shu-Shih Y. Wu modeled a spy network as a graph and then defined the neighbor integrity of a graph to obtain the vulnerability of a spy network [10]. The neighbor integrity of a graph G, is defined to be [Formula: see text], where S is any vertex subversion strategy of G and c(G/S) is the maximum order of the components of G/S. In this paper, we investigate the transformation graphs G-+-, G+--, G++-, G---, G+-+, G-++, G--+ and G+++ of a graph G, and determine their neighbor integrity.

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