A Note on Conditional Edge Connectivity of Hypercube Networks

2021 ◽  
pp. 2150007
Author(s):  
J. B. Saraf ◽  
Y. M. Borse
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Hypercube Networks ◽  
Sharp Lower Bound

Let [Formula: see text] be a connected graph with minimum degree at least [Formula: see text] and let [Formula: see text] be an integer such that [Formula: see text] The conditional [Formula: see text]-edge ([Formula: see text]-vertex) cut of [Formula: see text] is defined as a set [Formula: see text] of edges (vertices) of [Formula: see text] whose removal disconnects [Formula: see text] leaving behind components of minimum degree at least [Formula: see text] The characterization of a minimum [Formula: see text]-vertex cut of the [Formula: see text]-dimensional hypercube [Formula: see text] is known. In this paper, we characterize a minimum [Formula: see text]-edge cut of [Formula: see text] Also, we obtain a sharp lower bound on the number of vertices of an [Formula: see text]-edge cut of [Formula: see text] and obtain some consequences.

ON EDGE-CONNECTIVITY AND SUPER EDGE-CONNECTIVITY OF INTERCONNECTION NETWORKS MODELED BY PRODUCT GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 143-150
Author(s):  
CHUNXIANG WANG
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Product Graph ◽  
Product Graphs

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G–F contains at least two vertices. Let two connected graphs Gm and Gp have m and p vertices, minimum degree δ(Gm) and δ(Gp), edge-connectivity λ(Gm) and λ(Gp), respectively. This paper shows that min {pλ(Gm), λ(Gp) + δ(Gm), δ(Gm)(λ(Gp) + 1), (δ(Gm) + 1)λ(Gp)} ≤ λ(Gm * Gp) ≤ δ(Gm) + δ(Gp), where the product graph Gm * Gp of two given graphs Gm and Gp, defined by J. C. Bermond et al. [J. Combin. Theory B36 (1984) 32–48] in the context of the so-called (△, D)-problem, is one interesting model in the design of large reliable networks. Moreover, this paper determines λ′(Gm * Gp) ≤ min {pδ(Gm), ξ(Gp) + 2δ(Gm)} and λ′(G1 ⊕ G2) ≥ min {n, λ1 + λ2} if δ1 = δ2.

scholarly journals $\delta$-Connectivity in Random Lifts of Graphs

10.37236/6639 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Shashwat Silas
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Wreath Product ◽  
Connected Graph ◽  
Minimum Degree ◽  
Symmetric Groups ◽  
Gamma Delta ◽  
Random Generators

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least  $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

Maximal matchings in graphs with given minimal and maximal degrees

1976 ◽  
Vol 79 (2) ◽  
pp. 221-234 ◽  
Author(s):  
B. Bollobás ◽  
S. E. Eldridge
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Minimal Degree ◽  
Maximal Degree ◽  
Slight Extension

In this note we determine the greatest lower bound of the number of independent edges in a graph in terms of the number of vertices, the minimal degree and the maximal degree. More generally we examine the greatest lower bound of the number of independent edges in a K-connected (or λ-edge-connected) graph with given number of vertices and given minimal and maximal degrees. These results extend those of Erdös and Pósa (3) and Weinstein (7). Our proof makes heavy use of Berge's slight extension of Tutte's characterization of graphs with 1-factors.

scholarly journals A sharp lower bound of the spectral radius of simple graphs

2009 ◽  
Vol 3 (2) ◽  
pp. 379-385 ◽  
Author(s):  
Shengbiao Hu
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Connected Graph ◽  
Simple Graphs ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Two Parameter ◽  
Sharp Lower Bound

Let G be a simple connected graph with n vertices and let p(G) be its spectral radius. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i. Let Ni = ?j~i tj and Mi = ?j~i Nj. We find a sharp lower bound of p(G), which only contains two parameter Ni and Mi. Our result extends recent known results.

scholarly journals A Sharp Improvement of a Theorem of Bauer and Schmeichel

2018 ◽  
pp. 15-34
Author(s):  
Zhora Nikoghosyan
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Classical Result ◽  
Minimum Degree

Let G be a graph on n vertices with minimum degree δ. The earliest nontrivial lower bound for the circumference c (the length of a longest cycle in G) was established in 1952 due to Dirac in terms of n and δ: (i) if G is a 2-connected graph, then c ≥ min{n, 2δ}. The bound in Theorem (i) is sharp. In 1986, Bauer and Schmeichel gave a version of this classical result for 1-tough graphs: (ii) if G is a 1-tough graph, then c ≥ min{n, 2δ + 2}. In this paper we present an improvement of (ii), which is sharp for each n: (iii) if G is a 1-tough graph, then c ≥ min{n, 2δ + 2} when n ≡ 1(mod 3); c ≥ min{n, 2δ + 3} when n ≡ 2(mod 3) or n ≡ 1(mod 4); and c ≥ min{n, 2δ + 4} otherwise.

SUPER EDGE CONNECTIVITY OF KRONECKER PRODUCTS OF GRAPHS

2014 ◽  
Vol 25 (01) ◽  
pp. 59-65 ◽  
Author(s):  
XIANG-LAN CAO ◽  
ELKIN VUMAR
Keyword(s):  
Connected Graph ◽  
Kronecker Product ◽  
Minimum Degree ◽  
Kronecker Products ◽  
Edge Connectivity ◽  
Vertex Set ◽  
Product Of Graphs

A graph G is said to be super edge connected (in short super – λ) if every minimum edge cut isolates a vertex of G. The Kronecker product of graphs G and H is the graph with vertex set V(G × H) = V(G) × V(H), where two vertices (u1, v1) and (u2, v2) are adjacent in G × H if u1u2 ∈ E(G) and v1v2 ∈ E(H). Let G be a connected graph, and let δ(G) and λ(G) be the minimum degree and the edge-connectivity of G, respectively. In this paper we prove that G × Kn is super-λ for n ≥ 3, if λ(G) = δ(G) and G ≇ K2. Furthermore, we show that K2 × Kn is super-λ when n ≥ 4. Similar results for G × Tn are also obtained, where Tn is the graph obtained from Kn by adding a loop to every vertex of Kn.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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