5,556 research outputs found
Some Implications on Amorphic Association Schemes
AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition
Product Dimension of Forests and Bounded Treewidth Graphs
The product dimension of a graph G is defined as the minimum natural number l
such that G is an induced subgraph of a direct product of l complete graphs. In
this paper we study the product dimension of forests, bounded treewidth graphs
and k-degenerate graphs. We show that every forest on n vertices has a product
dimension at most 1.441logn+3. This improves the best known upper bound of
3logn for the same due to Poljak and Pultr. The technique used in arriving at
the above bound is extended and combined with a result on existence of
orthogonal Latin squares to show that every graph on n vertices with a
treewidth at most t has a product dimension at most (t+2)(logn+1). We also show
that every k-degenerate graph on n vertices has a product dimension at most
\ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by
Eaton and Rodl.Comment: 12 pages, 3 figure
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
An Upper Bound for the Representation Number of Graphs with Fixed Order
A graph has a representation modulo n if there exists an injective map f: {V(G)} -\u3e {0, 1, ... , n @ 1} such that vertices u and v are adjacent if and only if |f(u)@f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erdos and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible (Refer to PDF file for exact formulas)
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