118 research outputs found
Counterexamples to conjectures on graph distance measures based on topological indexes
In this paper we disprove three conjectures from [M. Dehmer, F.
Emmert-Streib, Y. Shi, Interrelations of graph distance measures based on
topological indices, PLoS ONE 9 (2014) e94985] on graph distance measures based
on topological indices by providing explicit classes of trees that do not
satisfy proposed inequalities. The constructions are based on the families of
trees that have the same Wiener index, graph energy or Randic index - but
different degree sequences.Comment: 9 pages, 2 figure
The asymptotic value of Randic index for trees
Let denote the set of all unrooted and unlabeled trees with
vertices, and a double-star. By assuming that every tree of
is equally likely, we show that the limiting distribution of
the number of occurrences of the double-star in is
normal. Based on this result, we obtain the asymptotic value of Randi\'c index
for trees. Fajtlowicz conjectured that for any connected graph the Randi\'c
index is at least the average distance. Using this asymptotic value, we show
that this conjecture is true not only for almost all connected graphs but also
for almost all trees.Comment: 12 page
A survey of recent results in (generalized) graph entropies
The entropy of a graph was first introduced by Rashevsky \cite{Rashevsky} and
Trucco \cite{Trucco} to interpret as the structural information content of the
graph and serve as a complexity measure. In this paper, we first state a number
of definitions of graph entropy measures and generalized graph entropies. Then
we survey the known results about them from the following three respects:
inequalities and extremal properties on graph entropies, relationships between
graph structures, graph energies, topological indices and generalized graph
entropies, complexity for calculation of graph entropies. Various applications
of graph entropies together with some open problems and conjectures are also
presented for further research.Comment: This will appear as a chapter "Graph Entropy: Recent Results and
Perspectives" in a book: Mathematical Foundations and Applications of Graph
Entrop
Degree based Topological indices of Hanoi Graph
There are various topological indices for example distance based topological
indices and degree based topological indices etc. In QSAR/QSPR study,
physiochemical properties and topological indices for example atom bond
connectivity index, fourth atom bond connectivity index, Randic connectivity
index, sum connectivity index, and so forth are used to characterize the
chemical compound. In this paper we computed the edge version of atom bond
connectivity index, fourth atom bond connectivity index, Randic connectivity
index, sum connectivity index, geometric-arithmetic index and fifth
geometric-arithmetic index of Double-wheel graph and Hanoi graph. The results
are analyzed and the general formulas are derived for the above mentioned
families of graphs
Bounds and power means for the general Randic index
We review bounds for the general Randi\'c index, , and use the power mean inequality to prove, for example,
that for , where is the
spectral radius of a graph. This enables us to strengthen various known lower
and upper bounds for and to generalise a non-spectral bound due to
Bollob\'as \emph{et al}. We also prove that the zeroth-order general Randi\'c
index, for
Paths and animals in unbounded degree graphs with repulsion
A class of countable infinite graphs with unbounded vertex degree is
considered. In these graphs, the vertices of large degree `repel' each other,
which means that the path distance between two such vertices cannot be smaller
than a certain function of their degrees. Assuming that this function increases
sufficiently fast, we prove that the number of finite connected subgraphs
(animals) of order N containing a given vertex x is exponentially bounded in N
for N belonging to an infinite subset N_x of natural numbers. Under a less
restrictive condition, the same result is obtained for the number of simple
paths originated at a given vertex. These results are then applied to a number
of problems, including estimating the growth of the Randi\'c index and of the
number of greedy animals
Short note on Randi\'{c} energy
In this paper, we consider the Randi\'{c} energy of simple connected
graphs. We provide upper bounds for in terms of the number of vertices and
the nullity of the graph. We present families of graphs that satisfy the
Conjecture proposed by Gutman, Furtula and Bozkurt \cite{Gutman1} about the
maximal RE. For example, we show that starlikes of odd order satisfy the
conjecture.Comment: 14 page
Further results on degree based topological indices of certain chemical networks
There are various topological indices such as degree based topological
indices, distance based topological indices and counting related topological
indices etc. These topological indices correlate certain physicochemical
properties such as boiling point, stability of chemical compounds. In this
paper, we compute the sum-connectivity index and multiplicative Zagreb indices
for certain networks of chemical importance like silicate networks, hexagonal
networks, oxide networks, and honeycomb networks. Moreover, a comparative study
using computer-based graphs has been made to clarify their nature for these
families of networks.Comment: Submitte
Some physical and chemical indices of clique-inserted-lattices
The operation of replacing every vertex of an -regular lattice by a
complete graph of order is called clique-inserting, and the resulting
lattice is called the clique-inserted-lattice of . For any given -regular
lattice, applying this operation iteratively, an infinite family of -regular
lattices is generated. Some interesting lattices including the 3-12-12 lattice
can be constructed this way. In this paper, we reveal the relationship between
the energy and resistance distance of an -regular lattice and that of its
clique-inserted-lattice. As an application, the asymptotic energy per vertex
and average resistance distance of the 3-12-12 and 3-6-24 lattices are
computed. We also give formulae expressing the numbers of spanning trees and
dimers of the -th iterated clique-inserted lattices in terms of that of the
original lattice. Moreover, we show that new families of expander graphs can be
constructed from the known ones by clique-inserting
Degree sequence of the generalized Sierpinski graph
We determine the degree sequence of the generalized Sierpinski graph and its
general first Zagreb index in terms of the same parameters of the base graph G
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