71 research outputs found
On the unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial
of a graph, including unimodality, log-concavity and reality of zeros. We
establish recurrence relations and give factorizations of independence
polynomials for certain classes of graphs. As applications we settle some
unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric
k-Colorability is Graph Automaton Recognizable
Automata operating on general graphs have been introduced by virtue of
graphoids. In this paper we construct a graph automaton that recognizes
-colorable graphs
On Some Closure Properties of nc-eNCE Graph Grammars
In the study of automata and grammars, closure properties of the associated
languages have been studied extensively. In particular, closure properties of
various types of graph grammars have been examined in (Rozenberg and Welzl,
Inf. and Control,1986) and (Rozenberg and Welzl, Acta Informatica,1986). In
this paper we examine some critical closure properties of the nc-eNCE graph
grammars discussed in (Jayakrishna and Mathew, Symmetry 2023) and (Jayakrishna
and Mathew, ICMICDS 2022).Comment: 14 pages,9 figures, to be submitted to Theory of Computin
Forbidden Subgraphs in Connected Graphs
Given a set of connected non acyclic graphs, a
-free graph is one which does not contain any member of as copy.
Define the excess of a graph as the difference between its number of edges and
its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating
function (EGF for brief) of connected -free graphs of excess equal to
(). For each fixed , a fundamental differential recurrence
satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to
solve this nonlinear recurrence for the first few values of by means of
graph surgery. We also show that for any finite collection of non-acyclic
graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the
generating function, , of Cayley's rooted (non-planar) labelled trees. From
this, we prove that almost all connected graphs with nodes and edges
are -free, whenever and by means of
Wright's inequalities and saddle point method. Limiting distributions are
derived for sparse connected -free components that are present when a
random graph on nodes has approximately edges. In particular,
the probability distribution that it consists of trees, unicyclic components,
, -cyclic components all -free is derived. Similar results are
also obtained for multigraphs, which are graphs where self-loops and
multiple-edges are allowed
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
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