28 research outputs found
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
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page
The inertia of unicyclic graphs and the implications for closed-shells
AbstractThe inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs
No mixed graph with the nullity
A mixed graph is obtained from a simple undirected graph ,
the underlying graph of , by orienting some edges of . Let
be the cyclomatic number of with
the number of connected components of , be the matching number of
, and be the nullity of . Chen et al.
(2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that
,
respectively, and they characterized the mixed graphs with nullity attaining
the upper bound and the lower bound. In this paper, we prove that there is no
mixed graph with nullity . Moreover,
for fixed , there are infinitely many connected mixed graphs with nullity
is proved