scholarly journals Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs

2018 ◽  
Vol 3 (2) ◽  
pp. 419-426 ◽  
Author(s):  
Donglin Liu ◽  
Chunxiang Wang ◽  
Shaohui Wang
Keyword(s):  
Interconnection Networks ◽  
Sufficient Conditions ◽  
Product Graph ◽  
Hamiltonian Connected ◽  
Product Graphs ◽  
Product Of Graphs ◽  
Theory Series

AbstractThe product graph Gm *Gp of two given graphs Gm and Gp, defined by J.C. Bermond et al.[J Combin Theory, Series B 36(1984) 32-48] in the context of the so-called (Δ,D)-problem, is one interesting model in the design of large reliable networks. This work deals with sufficient conditions that guarantee these product graphs to be hamiltonian-connected. Moreover, we state product graphs for which provide panconnectivity of interconnection networks modeled by a product of graphs with faulty elements.

On the Hamilton Cycle of the Hypercube

2011 ◽  
Vol 480-481 ◽  
pp. 922-927 ◽  
Author(s):  
Yan Zhong Hu ◽  
Hua Dong Wang
Keyword(s):  
Euler Characteristic ◽  
Interconnection Networks ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
The Other ◽  
Product Graph ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Isomorphic Graphs ◽  
The Relationship

Hypercube is one of the basic types of interconnection networks. In this paper, we use the concept of the Cartesian product graph to define the hypercube Qn, we study the relationship between the isomorphic graphs and the Cartesian product graphs, and we get the result that there exists a Hamilton cycle in the hypercube Qn. Meanwhile, the other properties of the hypercube Qn, such as Euler characteristic and bipartite characteristic are also introduced.

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 On very strongly perfect Cartesian product graphs

2021 ◽  
Vol 7 (2) ◽  
pp. 2634-2645
Author(s):  
Ganesh Gandal ◽  
◽  
R Mary Jeya Jothi ◽  
Narayan Phadatare ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Finite Graphs ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Necessary And Sufficient ◽  
And Control ◽  
Product Of Graphs ◽  
Real Time Application

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

scholarly journals q-Titchmarsh-Weyl theory: series expansion

2012 ◽  
Vol 205 ◽  
pp. 67-118
Author(s):  
M. H. Annaby ◽  
Z. S. Mansour ◽  
I. A. Soliman
Keyword(s):  
Limit Point ◽  
Eigenfunction Expansion ◽  
Bessel Functions ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Worked Example ◽  
Weyl Theory ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Theory Series

AbstractWe establish aq-Titchmarsh-Weyl theory for singularq-Sturm-Liouville problems. We defineq-limit-point andq-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jacksonq-Bessel functions is given. This example leads to the completeness of a wide class ofq-cylindrical functions.

On degree-based topological descriptors of strong product graphs

2016 ◽  
Vol 94 (6) ◽  
pp. 559-565 ◽  
Author(s):  
Shehnaz Akhter ◽  
Muhammad Imran
Keyword(s):  
Physicochemical Properties ◽  
Strain Energy ◽  
Chemical Compounds ◽  
Topological Indices ◽  
Topological Descriptors ◽  
Zagreb Indices ◽  
Strong Product ◽  
Product Graphs ◽  
Product Of Graphs ◽  
Atom Bond

Topological descriptors are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as Randić, atom–bond connectivity, and geometric–arithmetic are used to predict the bioactivity of different chemical compounds. There are certain types of topological descriptors such as degree-based topological indices, distance-based topological indices, counting-related topological indices, etc. Among degree-based topological indices, the so-called atom–bond connectivity and geometric–arithmetic are of vital importance. These topological indices correlate certain physicochemical properties such as boiling point, stability, strain energy, etc., of chemical compounds. In this paper, analytical closed formulas for Zagreb indices, multiplicative Zagreb indices, harmonic index, and sum-connectivity index of the strong product of graphs are determined.

On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

1998 ◽  
Vol 7 (4) ◽  
pp. 397-401 ◽  
Author(s):  
OLLE HÄGGSTRÖM
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Complete Graph ◽  
Continuous Time ◽  
Product Graph ◽  
Product Graphs ◽  
Continuous Time Random Walks ◽  
Discrete Time Case

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

scholarly journals Maximally and super connected multisplit graphs and digraphs

2019 ◽  
Vol 13 (1) ◽  
pp. 224-239
Author(s):  
Litao Guo ◽  
Guifu Su ◽  
Lutz Volkmann ◽  
Xingke Zhao
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Sufficient Conditions ◽  
Independent Set ◽  
Stable Sets ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
Vertex Set ◽  
Component Failures

Fault tolerance is especially important for interconnection networks, since the growing size of networks increases their vulnerability to component failures. A classical measure for the fault tolerance of a network in the case of vertex failures is its connectivities. A network based on a graph G = (X1?X2?...? Xk,I,E) is called a k-multisplit network, if its vertex set V can be partitioned into k+1 stable sets I,X1,X2,...,Xk such that X1 ?X2?...?Xk induces a complete k-partite graph and I is an independent set. In this note, we first show that: for any non-complete connected k-multisplit graph G = (X1?X2?...?Xk,I,E) with k ? 3 and |X1| ? |X2| ?...? |Xk|, each of the following holds (1) If |X1?X2?...?Xk-1| ? ?, then k(G) = ?(G). (2) If |X1?X2?...?Xk-1| < ?, then k(G)? |X1? X2?...? Xk-1|. (3) ?(G) = ?(G). (4) If |X1?X2?...?Xk-1| > ? with respect to |X1| ? 2 and ? ? 2, then G is super-k. (5) G is super-?. In addition, we present sufficient conditions for digraphs to be maximally edge-connected and super-edge connected in terms of the zeroth-order general Randic index of digraphs.

scholarly journals q-Titchmarsh-Weyl theory: series expansion

2012 ◽  
Vol 205 ◽  
pp. 67-118 ◽  
Author(s):  
M. H. Annaby ◽  
Z. S. Mansour ◽  
I. A. Soliman
Keyword(s):  
Limit Point ◽  
Eigenfunction Expansion ◽  
Bessel Functions ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Worked Example ◽  
Weyl Theory ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Theory Series

AbstractWe establish a q-Titchmarsh-Weyl theory for singular q-Sturm-Liouville problems. We define q-limit-point and q-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson q-Bessel functions is given. This example leads to the completeness of a wide class of q-cylindrical functions.

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