2 research outputs found
The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter
The conditional diameter of a connected graph is defined as
follows: given a property of a pair of
subgraphs of , the so-called \emph{conditional diameter} or -{\em diameter} measures the maximum distance among subgraphs satisfying
. That is, In this paper we consider the conditional diameter in
which requires that for all , for all , and for some integers and
, where denotes the degree of
a vertex of , denotes the minimum degree and the
maximum degree of . The conditional diameter obtained is called
-\emph{diameter}. We obtain upper bounds on the -diameter by using the -alternating polynomials on the mesh of
eigenvalues of an associated weighted graph. The method provides also bounds
for other parameters such as vertex separators
On the Randi\'{c} index and conditional parameters of a graph
The aim of this paper is to study some parameters of simple graphs related
with the degree of the vertices. So, our main tool is the matrix
whose ()-entry is where denotes the degree of the vertex . We study
the Randi\'{c} index and some interesting particular cases of conditional
excess, conditional Wiener index, and conditional diameter. In particular,
using the matrix or its eigenvalues, we obtain tight bounds on the
studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243