23,926 research outputs found
Split digraphs
We generalize the class of split graphs to the directed case and show that
these split digraphs can be identified from their degree sequences. The first
degree sequence characterization is an extension of the concept of splittance
to directed graphs, while the second characterization says a digraph is split
if and only if its degree sequence satisfies one of the Fulkerson inequalities
(which determine when an integer-pair sequence is digraphic) with equality.Comment: 14 pages, 2 figures; Accepted author manuscript (AAM) versio
Intersecting Families of Permutations
A set of permutations is said to be {\em k-intersecting} if
any two permutations in agree on at least points. We show that for any
, if is sufficiently large depending on , then the
largest -intersecting subsets of are cosets of stabilizers of
points, proving a conjecture of Deza and Frankl. We also prove a similar result
concerning -cross-intersecting subsets. Our proofs are based on eigenvalue
techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that
the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is
Theorem 27 for k > 1. An alternative proof of the equality part of the
Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and
2
Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States
We introduce a unifying and generalizing framework for complex and detailed
balanced steady states in chemical reaction network theory. To this end, we
generalize the graph commonly used to represent a reaction network.
Specifically, we introduce a graph, called a reaction graph, that has one edge
for each reaction but potentially multiple nodes for each complex. A special
class of steady states, called node balanced steady states, is naturally
associated with such a reaction graph. We show that complex and detailed
balanced steady states are special cases of node balanced steady states by
choosing appropriate reaction graphs. Further, we show that node balanced
steady states have properties analogous to complex balanced steady states, such
as uniqueness and asymptotical stability in each stoichiometric compatibility
class. Moreover, we associate an integer, called the deficiency, to a reaction
graph that gives the number of independent relations in the reaction rate
constants that need to be satisfied for a positive node balanced steady state
to exist.
The set of reaction graphs (modulo isomorphism) is equipped with a partial
order that has the complex balanced reaction graph as minimal element. We
relate this order to the deficiency and to the set of reaction rate constants
for which a positive node balanced steady state exists
Using Canonical Forms for Isomorphism Reduction in Graph-based Model Checking
Graph isomorphism checking can be used in graph-based model checking to achieve symmetry reduction. Instead of one-to-one comparing the graph representations of states, canonical forms of state graphs can be computed. These canonical forms can be used to store and compare states. However, computing a canonical form for a graph is computationally expensive. Whether computing a canonical representation for states and reducing the state space is more efficient than using canonical hashcodes for states and comparing states one-to-one is not a priori clear. In this paper these approaches to isomorphism reduction are described and a preliminary comparison is presented for checking isomorphism of pairs of graphs. An existing algorithm that does not compute a canonical form performs better that tools that do for graphs that are used in graph-based model checking. Computing canonical forms seems to scale better for larger graphs
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