1,489 research outputs found
The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter
The conditional diameter of a connected graph is defined as
follows: given a property of a pair of
subgraphs of , the so-called \emph{conditional diameter} or -{\em diameter} measures the maximum distance among subgraphs satisfying
. That is, In this paper we consider the conditional diameter in
which requires that for all , for all , and for some integers and
, where denotes the degree of
a vertex of , denotes the minimum degree and the
maximum degree of . The conditional diameter obtained is called
-\emph{diameter}. We obtain upper bounds on the -diameter by using the -alternating polynomials on the mesh of
eigenvalues of an associated weighted graph. The method provides also bounds
for other parameters such as vertex separators
Eigenvalue interlacing and weight parameters of graphs
Eigenvalue interlacing is a versatile technique for deriving results in
algebraic combinatorics. In particular, it has been successfully used for
proving a number of results about the relation between the (adjacency matrix or
Laplacian) spectrum of a graph and some of its properties. For instance, some
characterizations of regular partitions, and bounds for some parameters, such
as the independence and chromatic numbers, the diameter, the bandwidth, etc.,
have been obtained. For each parameter of a graph involving the cardinality of
some vertex sets, we can define its corresponding weight parameter by giving
some "weights" (that is, the entries of the positive eigenvector) to the
vertices and replacing cardinalities by square norms. The key point is that
such weights "regularize" the graph, and hence allow us to define a kind of
regular partition, called "pseudo-regular," intended for general graphs. Here
we show how to use interlacing for proving results about some weight parameters
and pseudo-regular partitions of a graph. For instance, generalizing a
well-known result of Lov\'asz, it is shown that the weight Shannon capacity
of a connected graph \G, with vertices and (adjacency matrix)
eigenvalues , satisfies \Theta\le
\Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where
is the (standard) Shannon capacity and \vecnu is the positive
eigenvector normalized to have smallest entry 1. In the special case of regular
graphs, the results obtained have some interesting corollaries, such as an
upper bound for some of the multiplicities of the eigenvalues of a
distance-regular graph. Finally, some results involving the Laplacian spectrum
are derived. spectrum are derived
On the Randi\'{c} index and conditional parameters of a graph
The aim of this paper is to study some parameters of simple graphs related
with the degree of the vertices. So, our main tool is the matrix
whose ()-entry is where denotes the degree of the vertex . We study
the Randi\'{c} index and some interesting particular cases of conditional
excess, conditional Wiener index, and conditional diameter. In particular,
using the matrix or its eigenvalues, we obtain tight bounds on the
studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243
The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs
Let Î be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ_1 > λ_2 > · · · > λ_d. For every k = 0,1, . . . ,d â1, the k-alternating polynomial P_k is defined to be the polynomial of degree k and norm |Peer Reviewe
Some applications of the proper and adjacency polynomials in the theory of graph spectra
Given a vertex u\inV of a graph , the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called -local spectrum of . These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for te distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here we develop the subject of these polynomials and gave a survey of some known results involving them. For instance, distance-regular graphs are characterized from their spectra and the number of vertices at ``extremal distance'' from each of their vertices. Afterwards, some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of and the weight -excess of a vertex. Given the integers , let denote the set of vertices which are at distance at least from a vertex , and there exist exactly (shortest) -paths from to each each of such vertices. As a main result, an upper bound for the cardinality of is derived, showing that decreases at least as , and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about -class association schemes, and prove some conjectures of Haemers and Van Dam about the number of vertices at distane three from every vertex of a regular graph with four distinct eigenvalues---setting and ---and, more generally, the number of non-adjacent vertices to every vertex , which have common neighbours with it.Peer Reviewe
Large Low-Diameter Graphs are Good Expanders
We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that "sufficiently large" graphs of fixed diameter and degree must be "good" expanders. We prove this statement for various definitions of "sufficiently large" (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research
Bounds on separated pairs of subgraphs, eigenvalues and related polynomials
We give a bound on the sizes of two sets of vertices at a given minimum distance (a separated pair of subgraphs) in a graph in terms of polynomials and the spectrum of the graph. We find properties of the polynomial optimizing the bound. Explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance are given, and we find graphs for which the bounds are tight.Graphs;Eigenvalues;Polynomials;mathematics
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