Tomoki Nakamigawa

CONNECTIVITY OF GENERALIZED HIERARCHICAL COMPLETELY-CONNECTED NETWORKS

As a network topology for a massively parallel computer system, Generalized Hierarchical Completely-Connected Networks (for short, HCC), which include conventional hierarchical networks, have been proposed. To apply the HCC to a parallel computer system effectively and to execute data processings on the HCC efficiently, the inherent fault-tolerant properties in HCC must be revealed. However, these properties have not been clarified enough. In this paper, node-connectivity is verified for HCC. Furthermore, the concept of block-connectivity related to node-connectivity of HCC is introduced, and fault-tolerance of HCC is discussed.

Enumeration Problems on the Expansion of a Chord Diagram

Abstract A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A pair of chords is called a crossing if the two chords intersect. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E \ S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E \ S ) ∪ { x 1 x 2 , x 3 x 4 } chord diagram E = E 1 ∪ E 2 is called complete bipartite of type ( m , n ), denoted by C m , n , if (1) both E 1 and E 2 are nonintersecting, (2) for every pair e 1 ∈ E 1 and e 2 ∈ E 2 , e 1 and e 2 are crossing, and (3) | E 1 | = m , | E 2 | = n . Let f m , n be the cardinality of the multiset of all nonintersecting chord diagrams generated from C m , n with a finite sequence of expansions. In this paper, it is shown ∑ m , n f m , n ( x m / m ! ) ( y n / n ! ) is 1 / ( cosh x cosh y − ( sinh x + sinh y ) ) .

Counting Lattice Paths via a New Cycle Lemma

Let

Node-disjoint paths in a level block of generalized hierarchical completely connected networks

\alpha,\beta,m,n

One-dimensional Tilings Using Tiles with Two Gap Lengths

be positive integers. Fix a line

The Expansion of a Chord Diagram and the Tutte Polynomial

L:y=\alpha x+\beta

An extremal problem for vertex partition of complete multipartite graphs

and a lattice point

Pebble exchange on graphs

Q=(m,n)

A ramsey-type theorem for multiple disjoint copies of induced subgraphs

on L. It is well known that the number of lattice paths from the origin to Q which touch L only at Q is given by

A Ramsey-type Theorem for Multiple Disjoint Copies of Induced Subgraphs (Extended Abstract)

\frac{\beta}{m+n}\binom{m+n}m.