2,094 research outputs found
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
The spectrum and toughness of regular graphs
In 1995, Brouwer proved that the toughness of a connected -regular graph
is at least , where is the maximum absolute value of
the non-trivial eigenvalues of . Brouwer conjectured that one can improve
this lower bound to and that many graphs (especially graphs
attaining equality in the Hoffman ratio bound for the independence number) have
toughness equal to . In this paper, we improve Brouwer's spectral
bound when the toughness is small and we determine the exact value of the
toughness for many strongly regular graphs attaining equality in the Hoffman
ratio bound such as Lattice graphs, Triangular graphs, complements of
Triangular graphs and complements of point-graphs of generalized quadrangles.
For all these graphs with the exception of the Petersen graph, we confirm
Brouwer's intuition by showing that the toughness equals ,
where is the smallest eigenvalue of the adjacency matrix of the
graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special
issue dedicated to the "Applications of Graph Spectra in Computer Science"
Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June
16-20, 201
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Steiner Distance in Product Networks
For a connected graph of order at least and , the
\emph{Steiner distance} among the vertices of is the minimum size
among all connected subgraphs whose vertex sets contain . Let and be
two integers with . Then the \emph{Steiner -eccentricity
} of a vertex of is defined by . Furthermore, the
\emph{Steiner -diameter} of is . In this paper, we investigate the Steiner distance and Steiner
-diameter of Cartesian and lexicographical product graphs. Also, we study
the Steiner -diameter of some networks.Comment: 29 pages, 4 figure
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