374 research outputs found
On Domination Number and Distance in Graphs
A vertex set of a graph is a \emph{dominating set} if each vertex of
either belongs to or is adjacent to a vertex in . The
\emph{domination number} of is the minimum cardinality of
as varies over all dominating sets of . It is known that , where denotes the diameter of . Define
as the largest constant such that for any vertices of an arbitrary connected graph ;
then in this view. The main result of this paper is that
for . It immediately follows that
, where and are
respectively the average distance and the Wiener index of of order . As
an application of our main result, we prove a conjecture of DeLaVi\~{n}a et
al.\;that , where denotes the
eccentricity of the boundary of an arbitrary connected graph .Comment: 5 pages, 2 figure
Domination in Functigraphs
Let and be disjoint copies of a graph , and let be a function. Then a \emph{functigraph}
has the vertex set and the edge set . A functigraph is a
generalization of a \emph{permutation graph} (also known as a \emph{generalized
prism}) in the sense of Chartrand and Harary. In this paper, we study
domination in functigraphs. Let denote the domination number of
. It is readily seen that . We
investigate for graphs generally, and for cycles in great detail, the functions
which achieve the upper and lower bounds, as well as the realization of the
intermediate values.Comment: 18 pages, 8 figure
Metric dimension and zero forcing number of two families of line graphs
summary:Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that for a simple and connected graph . Further, we show that when is a tree or when contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems
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