4,681 research outputs found
Counting invertible Schr\"odinger Operators over Finite Fields for Trees, Cycles and Complete Graphs
We count invertible Schr\"odinger operators (perturbations by diagonal
matrices of the adjacency matrix) over finite fieldsfor trees, cycles and
complete graphs.This is achieved for trees through the definition and use of
local invariants (algebraic constructions of perhapsindependent
interest).Cycles and complete graphs are treated by ad hoc methods.Comment: Final version to appear in Electronic Journal of Combinatoric
Faster Algorithms for the Maximum Common Subtree Isomorphism Problem
The maximum common subtree isomorphism problem asks for the largest possible
isomorphism between subtrees of two given input trees. This problem is a
natural restriction of the maximum common subgraph problem, which is -hard in general graphs. Confining to trees renders polynomial time
algorithms possible and is of fundamental importance for approaches on more
general graph classes. Various variants of this problem in trees have been
intensively studied. We consider the general case, where trees are neither
rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on
the mapped vertices and edges. For trees of order and maximum degree
our algorithm achieves a running time of by
exploiting the structure of the matching instances arising as subproblems. Thus
our algorithm outperforms the best previously known approaches. No faster
algorithm is possible for trees of bounded degree and for trees of unbounded
degree we show that a further reduction of the running time would directly
improve the best known approach to the assignment problem. Combining a
polynomial-delay algorithm for the enumeration of all maximum common subtree
isomorphisms with central ideas of our new algorithm leads to an improvement of
its running time from to ,
where is the order of the larger tree, is the number of different
solutions, and is the minimum of the maximum degrees of the input
trees. Our theoretical results are supplemented by an experimental evaluation
on synthetic and real-world instances
Cumulants of Hawkes point processes
We derive explicit, closed-form expressions for the cumulant densities of a
multivariate, self-exciting Hawkes point process, generalizing a result of
Hawkes in his earlier work on the covariance density and Bartlett spectrum of
such processes. To do this, we represent the Hawkes process in terms of a
Poisson cluster process and show how the cumulant density formulas can be
derived by enumerating all possible "family trees", representing complex
interactions between point events. We also consider the problem of computing
the integrated cumulants, characterizing the average measure of correlated
activity between events of different types, and derive the relevant equations.Comment: 11 pages, 4 figure
- …