1. CJM 2016 (vol 68 pp. 876)
 Ostrovskii, Mikhail; Randrianantoanina, Beata

Metric Spaces Admitting Lowdistortion Embeddings into All $n$dimensional Banach Spaces
For a fixed $K\gg 1$ and
$n\in\mathbb{N}$, $n\gg 1$, we study metric
spaces which admit embeddings with distortion $\le K$ into each
$n$dimensional Banach space. Classical examples include spaces
embeddable
into $\log n$dimensional Euclidean spaces, and equilateral spaces.
We prove that good embeddability properties are preserved under
the operation of metric composition of metric spaces. In
particular, we prove that $n$point ultrametrics can be
embedded with uniformly bounded distortions into arbitrary Banach
spaces of dimension $\log n$.
The main result of the paper is a new example of a family of
finite metric spaces which are not metric compositions of
classical examples and which do embed with uniformly bounded
distortion into any Banach space of dimension $n$. This partially
answers a question of G. Schechtman.
Keywords:basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametric Categories:46B85, 05C12, 30L05, 46B15, 52A21 

2. CJM 2016 (vol 68 pp. 655)
 Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad

Discrete Curvature and Abelian Groups
We study a natural discrete Bochnertype inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the socalled
$\Gamma_2$calculus (of BakryÃmery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest  particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Busertype inequalities
(Ã la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edgeisoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs  a result of independent
interest.
Keywords:Ricci curvature, graph theory, abelian groups Categories:53C21, 57M15 

3. CJM 2015 (vol 67 pp. 696)
 Zhang, Tong

Geography of Irregular Gorenstein 3folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3folds of general type. We generalize the classical NoetherCastelnuovo type inequalities for irregular surfaces to irregular 3folds according to the Albanese dimension.
Keywords:3fold, geography, irregular variety Category:14J30 

4. CJM 2014 (vol 67 pp. 721)
 Allen, Peter; Böttcher, Julia; Hladký, Jan; Piguet, Diana

A Density CorrÃ¡diHajnal 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 $k+1$ vertexdisjoint triangles.
This extends a result of Moon [Canad. J. Math. 20 (1968), 96102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the CorrÃ¡diHajnal Theorem.
Keywords:graph theory, Turan's Theorem, Mantel's Theorem, CorrÃ¡diHajnal Theorem, triangle Category:05C35 

5. CJM 2013 (vol 66 pp. 596)
 Eilers, Søren; Restorff, Gunnar; Ruiz, Efren

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 nonunital 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 

6. CJM 2011 (vol 64 pp. 102)
 Ishii, Atsushi; Iwakiri, Masahide

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 handlebodylinks 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 handlebodylinks.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph Categories:57M27, 57M15, 57M25 

7. CJM 2009 (vol 61 pp. 1239)
 Davidson, Kenneth R.; Yang, Dilian

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 

8. CJM 2008 (vol 60 pp. 1267)
 Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu

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 

9. CJM 2008 (vol 60 pp. 457)
 Teplyaev, Alexander

Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are
generalizations of postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically 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, selfsimilarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 

10. CJM 2008 (vol 60 pp. 64)
11. CJM 2007 (vol 59 pp. 828)
 Ortner, Ronald; Woess, Wolfgang

NonBacktracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Nonbacktracking 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 nonbacktracking 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 nonregular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
nonamenable if and only if the nonbacktracking $n$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 

12. CJM 2007 (vol 59 pp. 225)
13. CJM 2006 (vol 58 pp. 1268)
 Sims, Aidan

GaugeInvariant Ideals in the $C^*$Algebras of Finitely Aligned HigherRank Graphs
We produce a complete description of the lattice of gaugeinvariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gaugeinvariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the KirchbergPhillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$algebra Category:46L05 

14. CJM 2004 (vol 56 pp. 1022)
 Matignon, D.; Sayari, N.

NonOrientable 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 nonorientable 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 $q\leq 43\chi(\ch S)$ or $K$ is cabled and $q\leq 85\chi(\ch S)$.
As consequence, if $K$ is hyperbolic and $\chi(\ch S)=1$, then $q\leq 7$.
Keywords:Nonorientable surface, Dehn surgery, Intersection graphs Categories:57M25, 57N10, 57M15 

15. CJM 2002 (vol 54 pp. 795)
 Möller, Rögnvaldur G.

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 selfcontained 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 

16. CJM 2000 (vol 52 pp. 1057)
 Urakawa, Hajime

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 

17. CJM 1999 (vol 51 pp. 250)
 Combari, C.; Poliquin, R.; Thibault, L.

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\'eKuratowski
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 secondorder derivability of
convexly composite functions.
Keywords:epiconvergence, Mosco convergence, PainlevÃ©Kuratowski convergence, primallowernice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 
