9,525 research outputs found
P5-free augmenting graphs and the maximum stable set problem
AbstractThe complexity status of the maximum stable set problem in the class of P5-free graphs is unknown. In this paper, we first propose a characterization of all connected P5-free augmenting graphs. We then use this characterization to detect families of subclasses of P5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes
On a conjecture about tricyclic graphs with maximal energy
For a given simple graph , the energy of , denoted by , is defined as the sum of the absolute values of all eigenvalues of its
adjacency matrix, which was defined by I. Gutman. The problem on determining
the maximal energy tends to be complicated for a given class of graphs. There
are many approaches on the maximal energy of trees, unicyclic graphs and
bicyclic graphs, respectively. Let denote the graph with vertices obtained from three copies of and a path by
adding a single edge between each of two copies of to one endpoint of the
path and a single edge from the third to the other endpoint of the
. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P.
Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it
Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following
conjecture: Let be a tricyclic graphs on vertices with or
, then with equality
if and only if . Let denote the set of all
connected bipartite tricyclic graphs on vertices with three vertex-disjoint
cycles , and , where . In this paper, we try to
prove that the conjecture is true for graphs in the class ,
but as a consequence we can only show that this is true for most of the graphs
in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure
On Middle Cube Graphs
We study a family of graphs related to the -cube. The middle cube graph of parameter k is the subgraph of induced by the set of vertices whose binary representation has either or number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors)
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