143,070 research outputs found
On a conjecture about tricyclic graphs with maximal energy
For a given simple graph , the energy of , denoted by , is defined as the sum of the absolute values of all eigenvalues of its
adjacency matrix, which was defined by I. Gutman. The problem on determining
the maximal energy tends to be complicated for a given class of graphs. There
are many approaches on the maximal energy of trees, unicyclic graphs and
bicyclic graphs, respectively. Let denote the graph with vertices obtained from three copies of and a path by
adding a single edge between each of two copies of to one endpoint of the
path and a single edge from the third to the other endpoint of the
. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P.
Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it
Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following
conjecture: Let be a tricyclic graphs on vertices with or
, then with equality
if and only if . Let denote the set of all
connected bipartite tricyclic graphs on vertices with three vertex-disjoint
cycles , and , where . In this paper, we try to
prove that the conjecture is true for graphs in the class ,
but as a consequence we can only show that this is true for most of the graphs
in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure
Mathematical Aspects of Vacuum Energy on Quantum Graphs
We use quantum graphs as a model to study various mathematical aspects of the
vacuum energy, such as convergence of periodic path expansions, consistency
among different methods (trace formulae versus method of images) and the
possible connection with the underlying classical dynamics.
We derive an expansion for the vacuum energy in terms of periodic paths on
the graph and prove its convergence and smooth dependence on the bond lengths
of the graph. For an important special case of graphs with equal bond lengths,
we derive a simpler explicit formula.
The main results are derived using the trace formula. We also discuss an
alternative approach using the method of images and prove that the results are
consistent. This may have important consequences for other systems, since the
method of images, unlike the trace formula, includes a sum over special
``bounce paths''. We succeed in showing that in our model bounce paths do not
contribute to the vacuum energy. Finally, we discuss the proposed possible link
between the magnitude of the vacuum energy and the type (chaotic vs.
integrable) of the underlying classical dynamics. Within a random matrix model
we calculate the variance of the vacuum energy over several ensembles and find
evidence that the level repulsion leads to suppression of the vacuum energy.Comment: Fixed several typos, explain the use of random matrices in Section
Solution to a conjecture on the maximal energy of bipartite bicyclic graphs
The energy of a simple graph , denoted by , is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Let
denote the cycle of order and the graph obtained from joining
two cycles by a path with its two leaves. Let
denote the class of all bipartite bicyclic graphs but not the graph ,
which is obtained from joining two cycles and ( and ) by an edge. In [I. Gutman, D.
Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computer
experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and
Vidovi\'{c} conjectured that the bicyclic graph with maximal energy is
, for and . In [X. Li, J. Zhang, On bicyclic graphs
with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and
Zhang showed that the conjecture is true for graphs in the class
. However, they could not determine which of the two graphs
and has the maximal value of energy. In [B. Furtula, S.
Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy,
{\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations
up to were reported, supporting the conjecture. So, it is still
necessary to have a mathematical proof to this conjecture. This paper is to
show that the energy of is larger than that of , which
proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic
graphs, the conjecture is still open.Comment: 9 page
More on a Conjecture about Tricyclic Graphs with Maximal Energy *
Abstract The energy E(G) of a simple graph G is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. This concept was introduced by I. Gutman in 1977. Recently, Aouchiche et al. proposed a conjecture about tricyclic graphs: If G is a tricyclic graphs on n vertices with n = 20 or n ≥ 22, then E(G) ≤ E(P 6,6,6 n ) with equality if and only if G ∼ = P 6,6,6 n , where P 6,6,6 n denotes the graph with n ≥ 20 vertices obtained from three copies of C 6 and a path P n−18 by adding a single edge between each of two copies of C 6 to one endpoint of the path and a single edge from the third C 6 to the other endpoint of the P n−18 . Li et al. [X. Li, Y. Shi, M. Wei, J. Li, On a conjecture about tricyclic graphs with maximal energy, MATCH Commun. Math. Comput. Chem. 72 (2014) 183-214] proved that the conjecture is true for graphs in the graph class G(n; a, b, k), where G(n; a, b, k) denotes the set of all connected bipartite tricyclic graphs on n ≥ 20 vertices with three vertex-disjoint cycles C a , C b and C k , apart from 9 subclasses of such graphs. In this paper, we improve the above result and prove that apart from 7 smaller subclasses of such graphs the conjecture is true for graphs in the graph class G(n; a, b, k)
On Squared Distance Matrix of Complete Multipartite Graphs
Let be a complete -partite graph on
vertices. The distance between vertices and in
, denoted by is defined to be the length of the shortest path
between and . The squared distance matrix of is the
matrix with entry equal to if and equal to
if . We define the squared distance energy
of to be the sum of the absolute values of its eigenvalues. We determine
the inertia of and compute the squared distance energy
. More precisely, we prove that if for , then and if , then
Furthermore, we show that
for a fixed value of and , both the spectral radius of the squared
distance matrix and the squared distance energy of complete -partite graphs
on vertices are maximal for complete split graph and minimal for
Tur{\'a}n graph
Developments in the theory of randomized shortest paths with a comparison of graph node distances
There have lately been several suggestions for parametrized distances on a
graph that generalize the shortest path distance and the commute time or
resistance distance. The need for developing such distances has risen from the
observation that the above-mentioned common distances in many situations fail
to take into account the global structure of the graph. In this article, we
develop the theory of one family of graph node distances, known as the
randomized shortest path dissimilarity, which has its foundation in statistical
physics. We show that the randomized shortest path dissimilarity can be easily
computed in closed form for all pairs of nodes of a graph. Moreover, we come up
with a new definition of a distance measure that we call the free energy
distance. The free energy distance can be seen as an upgrade of the randomized
shortest path dissimilarity as it defines a metric, in addition to which it
satisfies the graph-geodetic property. The derivation and computation of the
free energy distance are also straightforward. We then make a comparison
between a set of generalized distances that interpolate between the shortest
path distance and the commute time, or resistance distance. This comparison
focuses on the applicability of the distances in graph node clustering and
classification. The comparison, in general, shows that the parametrized
distances perform well in the tasks. In particular, we see that the results
obtained with the free energy distance are among the best in all the
experiments.Comment: 30 pages, 4 figures, 3 table
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