1,848 research outputs found
A limit law of almost -partite graphs
For integers , we study (undirected) graphs with
vertices such that the vertices can be partitioned into parts
such that every vertex has at most neighbours in its own part. The set of
all such graphs is denoted \mbP_n(l,d). We prove a labelled first-order limit
law, i.e., for every first-order sentence , the proportion of graphs
in \mbP_n(l,d) that satisfy converges as . By
combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS}
we also prove that if are integers, then
\mb{Forb}(\mcK_{1, s_1, ..., s_l}) has a labelled first-order limit law,
where \mb{Forb}(\mcK_{1, s_1, ..., s_l}) denotes the set of all graphs with
vertices , for some , in which there is no subgraph isomorphic to
the complete -partite graph with parts of sizes . In
the course of doing this we also prove that there exists a first-order formula
(depending only on and ) such that the proportion of \mcG \in
\mbP_n(l,d) with the following property approaches 1 as : there
is a unique partition of into parts such that every vertex
has at most neighbours in its own part, and this partition, viewed as an
equivalence relation, is defined by
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
Collision Times in Multicolor Urn Models and Sequential Graph Coloring With Applications to Discrete Logarithms
Consider an urn model where at each step one of colors is sampled
according to some probability distribution and a ball of that color is placed
in an urn. The distribution of assigning balls to urns may depend on the color
of the ball. Collisions occur when a ball is placed in an urn which already
contains a ball of different color. Equivalently, this can be viewed as
sequentially coloring a complete -partite graph wherein a collision
corresponds to the appearance of a monochromatic edge. Using a Poisson
embedding technique, the limiting distribution of the first collision time is
determined and the possible limits are explicitly described. Joint distribution
of successive collision times and multi-fold collision times are also derived.
The results can be used to obtain the limiting distributions of running times
in various birthday problem based algorithms for solving the discrete logarithm
problem, generalizing previous results which only consider expected running
times. Asymptotic distributions of the time of appearance of a monochromatic
edge are also obtained for other graphs.Comment: Minor revision. 35 pages, 2 figures. To appear in Annals of Applied
Probabilit
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