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 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.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

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 Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Guisheng Jiang ◽  
Lifang Ren ◽  
Guidong Yu
Keyword(s):  
Minimum Degree ◽  
Sufficient Conditions ◽  
Wiener Index ◽  
Bipartite Graphs ◽  
Harary Index ◽  
Connected Graphs ◽  
Hamilton Connected

In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with δG≥t, where δG is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with δG≥t and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with δG≥t and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

scholarly journals Sufficient Conditions for Graphs to Be k -Connected, Maximally Connected, and Super-Connected

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Zhen-Mu Hong ◽  
Zheng-Jiang Xia ◽  
Fuyuan Chen ◽  
Lutz Volkmann
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Sufficient Conditions ◽  
Vertex Connectivity

Let G be a connected graph with minimum degree δ G and vertex-connectivity κ G . The graph G is k -connected if κ G ≥ k , maximally connected if κ G = δ G , and super-connected if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we present sufficient conditions for a graph with given minimum degree to be k -connected, maximally connected, or super-connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. Analogous results for triangle-free graphs with given minimum degree to be k -connected, maximally connected, or super-connected are also presented.

scholarly journals WHEN DO THREE LONGEST PATHS HAVE A COMMON VERTEX?

2009 ◽  
Vol 01 (01) ◽  
pp. 115-120 ◽  
Author(s):  
MARIA AXENOVICH
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Common Vertex ◽  
Outerplanar Graphs ◽  
Connected Graphs ◽  
Longest Paths

It is well known that any two longest paths in a connected graph share a vertex. It is also known that there are connected graphs where 7 longest paths do not share a common vertex. It was conjectured that any three longest paths in a connected graph have a vertex in common. In this note we prove the conjecture for outerplanar graphs and give sufficient conditions for the conjecture to hold in general.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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