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.

scholarly journals Wiener-type invariants on graph properties

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 489-502
Author(s):  
Qiannan Zhou ◽  
Ligong Wang ◽  
Yong Lu
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Graph Properties ◽  
Wiener Type ◽  
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 mainly give some sufficient conditions for a connected graph to be k-connected, ?-deficient, k-hamiltonian, k-edge-hamiltonian, k-path-coverable or satisfy ?(G)? k.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Sufficient Spectral Radius Conditions for Hamilton-Connectivity of k-Connected Graphs

2021 ◽  
Author(s):  
Qiannan Zhou ◽  
Hajo Broersma ◽  
Ligong Wang ◽  
Yong Lu
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Connected Graphs ◽  
Hamilton Connected

AbstractWe present two new sufficient conditions in terms of the spectral radius $$\rho (G)$$ ρ ( G ) guaranteeing that a k-connected graph G is Hamilton-connected, unless G belongs to a collection of exceptional graphs. We use the Bondy–Chvátal closure to characterize these exceptional graphs.

scholarly journals Comparison between Laplacian--energy--like invariant and Kirchhoff index

2016 ◽  
Vol 31 ◽  
pp. 27-41 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Hilal Ganie ◽  
Ivan Gutman
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Simple Connected Graph

For a simple connected graph G of order n, having Laplacian eigenvalues μ_1, μ_2, . . . ,μ_{n−1}, μ_n = 0, the Laplacian–energy–like invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = \sum_{i=1}^{n-1} \sqrt{μ_i} Kf(G) = \sum_{i=1}^{n-1} 1/μ_i, respectively. In this paper, LEL and Kf arecompared, and sufficient conditions for the inequality Kf(G) < LEL(G) are established.

scholarly journals Energy Conditions for Hamiltonicity of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Guidong Yu ◽  
Gaixiang Cai ◽  
Miaolin Ye ◽  
Jinde Cao
Keyword(s):  
Bipartite Graph ◽  
Adjacency Matrix ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Energy Conditions ◽  
Sufficient Condition ◽  
Hamilton Connected

LetGbe an undirected simple graph of ordern. LetA(G)be the adjacency matrix ofG, and letμ1(G)≤μ2(G)≤⋯≤μn(G)be its eigenvalues. The energy ofGis defined asℰ(G)=∑i=1n‍|μi(G)|. Denote byGBPTa bipartite graph. In this paper, we establish the sufficient conditions forGhaving a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement ofG, and give the sufficient condition forGBPThaving a Hamiltonian cycle in terms of the energy of the quasi-complement ofGBPT.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

HAMILTONIAN PROPERTIES OF FAULTY RECURSIVE CIRCULANT GRAPHS

2002 ◽  
Vol 03 (03n04) ◽  
pp. 273-289 ◽  
Author(s):  
CHANG-HSIUNG TSAI ◽  
JIMMY J. M. TAN ◽  
YEN-CHU CHUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Path ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Regular Node ◽  
Hamiltonian Properties

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

scholarly journals Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

2007 ◽  
Vol 3 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

scholarly journals Pelabelan 0-Anti Ajaib dan 2-Anti Ajaib untuk Graf Tangga L

2016 ◽  
Vol 12 (1) ◽  
pp. 63
Author(s):  
Quinoza Guvil ◽  
Roni Tri Putra
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Partition Dimension ◽  
Dimension Partition

For a connected graph  and a subset  of   . For a vertex  the distance betwen  and  is . For an ordered k-partition of ,  the representation of   with respect to  is    The k-partition  is a resolving partition if  are distinct for every  The minimum k for which there is a resolving partition of   is the partition dimension of   In this paper will shown resolving partition of  connected graph order  where  is a bipartite graph. Then it is shown dimension partition of bipartite graph, are pd(Kst)=n-1

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