881 research outputs found
The average solution of a TSP instance in a graph
We define the average -TSP distance of a graph as the
average length of a shortest walk visiting vertices, i.e. the expected
length of the solution for a random TSP instance with uniformly random
chosen vertices. We prove relations with the average -Steiner distance and
characterize the cases where equality occurs. We also give sharp bounds for
given the order of the graph.Comment: 9 pages, 3 figure
Maximum Wiener Indices of Unicyclic Graphs of Given Matching Number
In this article, we determine the maximum Wiener indices of unicyclic graphs
with given number of vertices and matching number. We also characterize the
extremal graphs. This solves an open problem of Du and Zhou.Comment: 14 pages, 9 figure
Five results on maximizing topological indices in graphs
In this paper, we prove a collection of results on graphical indices. We
determine the extremal graphs attaining the maximal generalized Wiener index
(e.g. the hyper-Wiener index) among all graphs with given matching number or
independence number. This generalizes some work of Dankelmann, as well as some
work of Chung. We also show alternative proofs for two recents results on
maximizing the Wiener index and external Wiener index by deriving it from
earlier results. We end with proving two conjectures. We prove that the maximum
for the difference of the Wiener index and the eccentricity is attained by the
path if the order is at least and that the maximum weighted Szeged
index of graphs of given order is attained by the balanced complete bipartite
graphs.Comment: 13 pages, 4 figure
Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs
In the original article ``Wiener index of Eulerian graphs'' [Discrete Applied
Mathematics Volume 162, 10 January 2014, Pages 247-250], the authors state that
the Wiener index (total distance) of an Eulerian graph is maximized by the
cycle. We explain that the initial proof contains a flaw and note that it is a
corollary of a result by Plesn\'ik, since an Eulerian graph is
-edge-connected. The same incorrect proof is used in two referencing papers,
``Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs'' [Bull.
Malays. Math. Sci. Soc. (2019) 42:67-78] and ``Harary index of Eulerian
graphs'' [J. Math. Chem., 59(5):1378-1394, 2021]. We give proofs of the main
results of those papers and the -edge-connected analogues.Comment: 5 Pages, 1 Figure Corrigendum of 3 papers, whose titles are combine
Counterexamples to conjectures on the occupancy fraction of graphs
The occupancy fraction of a graph is a (normalized) measure on the size of
independent sets under the hard-core model, depending on a variable (fugacity)
We present a criterion for finding the graph with minimum occupancy
fraction among graphs with a fixed order, and disprove five conjectures on the
extremes of the occupancy fraction and (normalized) independence polynomial for
certain graph classes of regular graphs with a given girth.Comment: 8 pages, 4 figure
The minimum number of maximal independent sets in twin-free graphs
The problem of determining the maximum number of maximal independent sets in
certain graph classes dates back to a paper of Miller and Muller and a question
of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be
less interesting due to simple examples such as stars. In this paper we show
that the problem becomes interesting when restricted to twin-free graphs, where
no two vertices have the same open neighbourhood. We consider the question for
arbitrary graphs, bipartite graphs and trees. The minimum number of maximal
independent sets turns out to be logarithmic in the number of vertices for
arbitrary graphs, linear for bipartite graphs and exponential for trees. In the
latter case, the minimum and the extremal graphs have been determined earlier
by Taletski\u{\i} and Malyshev, but we present a shorter proof.Comment: 17 pages, 7 figures, 5 table
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