6,865 research outputs found
An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
For a distance-regular graph with second largest eigenvalue (resp. smallest
eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where
equality only holds when the diameter equals two. Using this inequality we
study distance-regular graphs with fixed second largest eigenvalue.Comment: 15 pages, this is submitted to Linear Algebra and Applications
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
Broadcasting on Random Directed Acyclic Graphs
We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex , and
suppose all other vertices have indegree . Let the vertices at
distance from be called layer . At layer , is given a random
bit. At layer , each vertex receives bits from its parents in
layer , which are transmitted along independent binary symmetric channel
edges, and combines them using a -ary Boolean processing function. The goal
is to reconstruct with probability of error bounded away from using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For , and random DAGs with
layer sizes and majority processing functions, we identify the
critical threshold. For , we establish a similar result for NAND
processing functions. We also prove a partial converse for odd
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752
Taut distance-regular graphs and the subconstituent algebra
We consider a bipartite distance-regular graph with diameter at least
4 and valency at least 3. We obtain upper and lower bounds for the local
eigenvalues of in terms of the intersection numbers of and the
eigenvalues of . Fix a vertex of and let denote the corresponding
subconstituent algebra. We give a detailed description of those thin
irreducible -modules that have endpoint 2 and dimension . In an earlier
paper the first author defined what it means for to be taut. We obtain
three characterizations of the taut condition, each of which involves the local
eigenvalues or the thin irreducible -modules mentioned above.Comment: 29 page
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