31,704 research outputs found
Balanced and 1-balanced graph constructions
AbstractThere are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|βΟ(G)), where Ο(G) denotes the number of components of G. Graphs for which b(H)β€b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)β€g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced
Balanced Families of Perfect Hash Functions and Their Applications
The construction of perfect hash functions is a well-studied topic. In this
paper, this concept is generalized with the following definition. We say that a
family of functions from to is a -balanced -family
of perfect hash functions if for every , , the number
of functions that are 1-1 on is between and for some
constant . The standard definition of a family of perfect hash functions
requires that there will be at least one function that is 1-1 on , for each
of size . In the new notion of balanced families, we require the number
of 1-1 functions to be almost the same (taking to be close to 1) for
every such . Our main result is that for any constant , a
-balanced -family of perfect hash functions of size can be constructed in time .
Using the technique of color-coding we can apply our explicit constructions to
devise approximation algorithms for various counting problems in graphs. In
particular, we exhibit a deterministic polynomial time algorithm for
approximating both the number of simple paths of length and the number of
simple cycles of size for any
in a graph with vertices. The approximation is up to any fixed desirable
relative error
Nonbonding orbitals in fullerenes :β nuts and cores in singular polyhedral graphs
A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding Ο orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (β2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10β190β782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62β043 of the 398β383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies:β it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.peer-reviewe
Some results on uniform mixing on abelian Cayley graphs
In the past few decades, quantum algorithms have become a popular research
area of both mathematicians and engineers. Among them, uniform mixing provides
a uniform probability distribution of quantum information over time which
attracts a special attention. However, there are only a few known examples of
graphs which admit uniform mixing. In this paper, a characterization of abelian
Cayley graphs having uniform mixing is presented. Some concrete constructions
of such graphs are provided. Specifically, for cubelike graphs, it is shown
that the Cayley graph has uniform mixing if
the characteristic function of is bent. Moreover, a difference-balanced
property of the eigenvalues of an abelian Cayley graph having uniform mixing is
established. Furthermore, it is proved that an integral abelian Cayley graph
exhibits uniform mixing if and only if the underlying group is one of the
groups: , or
for some integers .
Thus the classification of integral abelian Cayley graphs having uniform mixing
is completed.Comment: 33 page
Irreducibility of configurations
In a paper from 1886, Martinetti enumerated small -configurations. One
of his tools was a construction that permits to produce a
-configuration from a -configuration. He called configurations
that were not constructible in this way irreducible configurations. According
to his definition, the irreducible configurations are Pappus' configuration and
four infinite families of configurations. In 2005, Boben defined a simpler and
more general definition of irreducibility, for which only two
-configurations, the Fano plane and Pappus' configuration, remained
irreducible. The present article gives a generalization of Boben's reduction
for both balanced and unbalanced -configurations, and proves several
general results on augmentability and reducibility. Motivation for this work is
found, for example, in the counting and enumeration of configurations
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
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