scholarly journals Sufficient conditions for hamiltonian properties of graphs

2019 ◽  
Author(s):  
Qiannan Zhou
Keyword(s):  
Sufficient Conditions ◽  
Hamiltonian Properties

The Hyper-Zagreb Index and Some Properties of Graphs

2020 ◽  
pp. 120-134 ◽  
Author(s):  
Rao Li
Keyword(s):  
Topological Index ◽  
Sufficient Conditions ◽  
Disjoint Paths ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Second Zagreb Index ◽  
Vertex Disjoint ◽  
Hamiltonian Properties

Let G = (V(G), E(G)) be a graph. The complement of G is denoted by Gc. The forgotten topological index of G, denoted F(G), is defined as the sum of the cubes of the degrees of all the vertices in G. The second Zagreb index of G, denoted M2(G), is defined as the sum of the products of the degrees of pairs of adjacent vertices in G. A graph Gisk-Hamiltonian if for all X ⊂V(G) with|X| ≤ k, the subgraph induced byV(G) - Xis Hamiltonian. Clearly, G is 0-Hamiltonian if and only if G is Hamiltonian. A graph Gisk-path-coverableifV(G) can be covered bykor fewer vertex-disjoint paths. Using F(Gc) and M2(Gc), Li obtained several sufficient conditions for Hamiltonian and traceable graphs (Rao Li, Topological Indexes and Some Hamiltonian Properties of Graphs). In this chapter, the author presents sufficient conditions based upon F(Gc) and M2(Gc)for k-Hamiltonian, k-edge-Hamiltonian, k-path-coverable, k-connected, and k-edge-connected graphs.

scholarly journals Wiener-type invariants and Hamiltonian properties of graphs

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4045-4058
Author(s):  
Qiannan Zhou ◽  
Ligong Wang ◽  
Yong Lu
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
General Graph ◽  
Wiener Type ◽  
Hamilton Connected ◽  
Hamiltonian Properties ◽  
Simple Connected Graph

The Wiener-type invariants of a simple connected graph G = (V(G), E(G)) can be expressed in terms of the quantities Wf = ? {u,v}?V(G)f(dG(u,v)) for various choices of the function f(x), where dG(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a bipartite graph to be Hamiltonian or a connected general graph to be Hamilton-connected and traceable from every vertex in terms of the Wiener-type invariants of G or the complement of G.

Graphs with cyclomatic number three having panconnected square

2017 ◽  
Vol 09 (05) ◽  
pp. 1750067
Author(s):  
G. L. Chia ◽  
W. Hemakul ◽  
S. Singhun
Keyword(s):  
Sufficient Conditions ◽  
Discrete Math ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Cyclomatic Number ◽  
Number Formula ◽  
Necessary And Sufficient ◽  
Hamiltonian Properties ◽  
Square Of A Graph

The square of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding edges joining those pairs of vertices whose distance from each other in [Formula: see text] is two. If [Formula: see text] is connected, then the cyclomatic number of [Formula: see text] is defined as [Formula: see text]. Graphs with cyclomatic number not more than [Formula: see text] whose square are panconnected have been characterized, among other things, in [G. L. Chia, S. H. Ong and L. Y. Tan, On graphs whose square have strong Hamiltonian properties, Discrete Math. 309 (2009) 4608–4613, G. L. Chia, W. Hemakul and S. Singhun, Graphs with cyclomatic number two having panconnected square, Discrete Math. 311 (2011) 850–855]. Here, we show that if [Formula: see text] has cyclomatic number [Formula: see text] and [Formula: see text] is panconnected, then [Formula: see text] is one of the eight families of graphs, [Formula: see text], defined in the paper. Further, we obtain necessary and sufficient conditions for three larger families of graphs (which contains [Formula: see text] as special cases) whose square are panconnected.

scholarly journals Improved sufficient conditions for Hamiltonian properties

2015 ◽  
Vol 35 (2) ◽  
pp. 329
Author(s):  
Jens-P. Bode ◽  
Anika Fricke ◽  
Arnfried Kemnitz
Keyword(s):  
Sufficient Conditions ◽  
Hamiltonian Properties

scholarly journals The Largest Eigenvalue and Some Hamiltonian Properties of Graphs

2018 ◽  
Vol 34 ◽  
pp. 389-392 ◽  
Author(s):  
Rao Li
Keyword(s):  
Sufficient Conditions ◽  
Largest Eigenvalue ◽  
Hamiltonian Properties ◽  
The Largest Eigenvalue

In this note, sufficient conditions, based on the largest eigenvalue, are presented for some Hamiltonian properties of graphs.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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