Expand all Collapse all | Results 1 - 14 of 14 |
1. CJM Online first
A density CorrÃ¡di-Hajnal Theorem We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles.
This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the CorrÃ¡di-Hajnal Theorem.
Keywords:graph theory, Turan's Theorem, Mantel's Theorem, CorrÃ¡di-Hajnal Theorem, triangle Category:05C35 |
2. CJM 2013 (vol 66 pp. 596)
The Ordered $K$-theory of a Full Extension Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has
the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal
of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the
corona factorization property. We prove that
$
0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0
$
is a full extension if and only if the extension is stenotic and
$K$-lexicographic. {As an immediate application, we extend the
classification result for graph $C^*$-algebras obtained by Tomforde
and the first named author to the general non-unital case. In
combination with recent results by Katsura, Tomforde, West and the
first author, our result may also be used to give a purely
$K$-theoretical description of when an essential extension of two
simple and stable graph $C^*$-algebras is again a graph
$C^*$-algebra.}
Keywords:classification, extensions, graph algebras Categories:46L80, 46L35, 46L05 |
3. CJM 2011 (vol 64 pp. 102)
Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies We introduce a flow of a spatial graph and see how invariants for
spatial graphs and handlebody-links are derived from those for flowed
spatial graphs.
We define a new quandle (co)homology by introducing a subcomplex of the
rack chain complex.
Then we define quandle colorings and quandle cocycle invariants for
spatial graphs and handlebody-links.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph Categories:57M27, 57M15, 57M25 |
4. CJM 2009 (vol 61 pp. 1239)
Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition
for higher rank graphs.
We present a detailed analysis of when this occurs
in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof
of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$.
The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown
that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq
\mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$
where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra.
Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectation Categories:47L55, 47L30, 47L75, 46L05 |
5. CJM 2008 (vol 60 pp. 1267)
Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves
for cryptographic use. For fast operations on these curves,
these papers also
initiated a study of the radix-$\tau$ expansion of integers in the number
fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window)
nonadjacent form of $\tau$-expansion of integers in
$\Q(\sqrt{-7})$ was first investigated by Solinas.
For integers in $\Q(\sqrt{-3})$, the nonadjacent form
and the window nonadjacent form of the $\tau$-expansion were
studied. These are used for efficient
point multiplications on Koblitz curves.
In this paper, we complete
the picture by producing the (window)
nonadjacent radix-$\tau$ expansions
for integers in all Euclidean imaginary quadratic number fields.
Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography Categories:11A63, 11R04, 11Y16, 11Y40, 14G50 |
6. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
7. CJM 2008 (vol 60 pp. 64)
Classification of Linear Weighted Graphs Up to Blowing-Up and Blowing-Down We classify linear weighted graphs up to the
blowing-up and blowing-down operations which are relevant for the
study of algebraic surfaces.
Keywords:weighted graph, dual graph, blowing-up, algebraic surface Categories:14J26, 14E07, 14R05, 05C99 |
8. CJM 2007 (vol 59 pp. 828)
Non-Backtracking Random Walks and Cogrowth of Graphs Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Non-backtracking random walk moves at each step with equal
probability to one of the ``forward'' neighbours of the actual state,
\emph{i.e.,} it does not go back along
the preceding edge to the preceding
state. This is not a Markov chain, but can be turned into a Markov
chain whose state space is the set of oriented edges of $X$. Thus we
obtain for infinite $X$ that the $n$-step non-backtracking transition
probabilities tend to zero, and we can also compute their limit when
$X$ is finite. This provides a short proof of old results concerning
cogrowth of groups, and makes the extension of that result to
arbitrary regular graphs rigorous. Even when $X$ is non-regular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
non-amenable if and only if the non-backtracking $n$-step transition
probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which
comprises the original cogrowth criterion for finitely generated
groups of Grigorchuk and Cohen.
Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability Categories:05C75, 60G50, 20F69 |
9. CJM 2007 (vol 59 pp. 225)
Harmonic Analysis on Metrized Graphs This paper studies the Laplacian operator on a metrized graph, and its
spectral theory.
Keywords:metrized graph, harmonic analysis, eigenfunction Categories:43A99, 58C40, 05C99 |
10. CJM 2006 (vol 58 pp. 1268)
Gauge-Invariant Ideals in the $C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gauge-invariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the Kirchberg--Phillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$-algebra Category:46L05 |
11. CJM 2004 (vol 56 pp. 1022)
Non-Orientable Surfaces and Dehn Surgeries Let $K$ be a knot in $S^3$. This paper is devoted to Dehn surgeries which create
$3$-manifolds containing a closed non-orientable surface $\ch S$. We look at the
slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the
surface and the intersection number $s$ between $\ch S$ and the core of the Dehn
surgery. We prove that if $\chi(\hat S) \geq 15 - 3q$, then $s=1$. Furthermore,
if $s=1$ then $q\leq 4-3\chi(\ch S)$ or $K$ is cabled and $q\leq 8-5\chi(\ch S)$.
As consequence, if $K$ is hyperbolic and $\chi(\ch S)=-1$, then $q\leq 7$.
Keywords:Non-orientable surface, Dehn surgery, Intersection graphs Categories:57M25, 57N10, 57M15 |
12. CJM 2002 (vol 54 pp. 795)
Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations Willis's structure theory of totally disconnected locally compact groups
is investigated in the context of permutation actions. This leads to new
interpretations of the basic concepts in the theory and also to new proofs
of the fundamental theorems and to several new results. The treatment of
Willis's theory is self-contained and full proofs are given of all the
fundamental results.
Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs Categories:22D05, 20B07, 20B27, 05C25 |
13. CJM 2000 (vol 52 pp. 1057)
The Spectrum of an Infinite Graph In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.
Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum Categories:05C50, 58G25 |
14. CJM 1999 (vol 51 pp. 250)
Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'e-Kuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized second-order derivability of
convexly composite functions.
Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 |