210 research outputs found
On the Core of a Unicyclic Graph
A set S is independent in a graph G if no two vertices from S are adjacent.
By core(G) we mean the intersection of all maximum independent sets. The
independence number alpha(G) is the cardinality of a maximum independent set,
while mu(G) is the size of a maximum matching in G. A connected graph having
only one cycle, say C, is a unicyclic graph. In this paper we prove that if G
is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G)
coincides with the union of cores of all trees in G-C.Comment: 8 pages, 5 figure
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
On the strong partition dimension of graphs
We present a different way to obtain generators of metric spaces having the
property that the ``position'' of every element of the space is uniquely
determined by the distances from the elements of the generators. Specifically
we introduce a generator based on a partition of the metric space into sets of
elements. The sets of the partition will work as the new elements which will
uniquely determine the position of each single element of the space. A set
of vertices of a connected graph strongly resolves two different vertices
if either or
, where . An ordered vertex partition of
a graph is a strong resolving partition for if every two different
vertices of belonging to the same set of the partition are strongly
resolved by some set of . A strong resolving partition of minimum
cardinality is called a strong partition basis and its cardinality the strong
partition dimension. In this article we introduce the concepts of strong
resolving partition and strong partition dimension and we begin with the study
of its mathematical properties. We give some realizability results for this
parameter and we also obtain tight bounds and closed formulae for the strong
metric dimension of several graphs.Comment: 16 page
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