187 research outputs found
Rainbow connection in -connected graphs
An edge-colored graph is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection
number of a connected graph , denoted by , is the smallest number of
colors that are needed in order to make rainbow connected. In this paper,
we proved that for all -connected graphs.Comment: 7 page
A Tur\'an-type problem on degree sequence
Given and a graph whose degree sequence is
, let . Caro and Yuster
introduced a Tur\'an-type problem for : given , how large can
be if has no subgraph of a particular type. Denote by
the maximum value of taken over all graphs with vertices that do
not contain as a subgraph. Clearly, , where
denotes the classical Tur\'an number, i.e., the maximum number of edges among
all -free graphs with vertices. Pikhurko and Taraz generalize this
Tur\'an-type problem: let be a non-negative increasing real function and
, and then define as the maximum value
of taken over all graphs with vertices that do not contain as
a subgraph. Observe that if ,
if . Bollob\'as and Nikiforov mentioned that it
is important to study concrete functions. They gave an example
, since counts the
-vertex subgraphs of with a dominating vertex.
Denote by the -partite Tur\'an graph of order . In this paper,
using the Bollob\'as--Nikiforov's methods, we give some results on
as follows: for ,
; for each , there exists a constant
such that for every and sufficiently large ,
; for a fixed -chromatic graph
and every , when is sufficiently large, we have
.Comment: 9 page
Complete solution to a problem on the maximal energy of unicyclic bipartite graphs
The energy of a simple graph , denoted by , is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Denote by
the cycle, and the unicyclic graph obtained by connecting a vertex of
with a leaf of \,. Caporossi et al. conjecture that the
unicyclic graph with maximal energy is for and .
In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it
Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that
is maximal within the class of the unicyclic bipartite -vertex
graphs differing from \,. And they also claimed that the energy of
and is quasi-order incomparable and left this as an open problem. In
this paper, by utilizing the Coulson integral formula and some knowledge of
real analysis, especially by employing certain combinatorial techniques, we
show that the energy of is greater than that of for
and , which completely solves this open problem and partially solves
the above conjecture.Comment: 8 page
On the maximal energy tree with two maximum degree vertices
For a simple graph , the energy is defined as the sum of the
absolute values of all eigenvalues of its adjacent matrix. For
and , denote by (or simply ) the tree formed from
a path on vertices by attaching 's on each end of the
path , and (or simply ) the tree formed from
by attaching 's on an end of the and
's on the vertex next to the end. In [X. Li, X. Yao, J. Zhang
and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math.
Chem. 45(2009), 962--973], Li et al. proved that among trees of order with
two vertices of maximum degree , the maximal energy tree is either the
graph or the graph , where . However, they
could not determine which one of and is the maximal energy tree.
This is because the quasi-order method is invalid for comparing their energies.
In this paper, we use a new method to determine the maximal energy tree. It
turns out that things are more complicated. We prove that the maximal energy
tree is for and any , while the maximal energy
tree is for and any . Moreover, for , the
maximal energy tree is for all but , for which is
the maximal energy tree. For , the maximal energy tree is for
all but is odd and , for which is the
maximal energy tree. For , the maximal energy tree is for all
but , for which is the maximal energy tree. One can
see that for most , is the maximal energy tree, is a
turning point, and and 4 are exceptional cases.Comment: 16 page
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