Kronecker Products and Local Joins of Graphs

Abstract

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

On the chromatic number of the product of graphs

“…We occasionally emphasize this last condition by calling cp a good n-coloring of G. We say G is n-chromatic and write x(G) = n if n is the minimum k for which there is a k-coloring of G . For [2,6,17,22].) Hell (121 observes for instance that (1) holds provided that a related topological space has a certain connectivity property; Greenwell and LOV~SZ [7J and Miller [lS] obtain results for products indexed by infinite sets.…”

New directions in modelling, analysis and design of WDM/OFDM-networks: (I) Optical Switching

“…There exist several nonisomorphic coverings, some of them are countable [29,30], though, in general, their computation is NP-complete [31]. Additionally, we may wish to increase the number of connections between the embedded prisms [32,33], in order to further reduce N S or implement expanders/concentrators. Such an attempts is similar to the multidimensional case in (22], though the implementation of the required 2x2-switches may be difficult.…”

Double covers of graphs

“…Our f i r s t type of double cover of G can now be dealt with using the fact that the category Giutuph. has products (see for example Farzan and Wailer [3] and the references therein). The product G A G of two graphs G. and G~ (often known as their Kronecker product) has vertexset V[G A C ) equal to the cartesian product v [G ) x V[GS\ of the vertex-sets of the given graphs, with adjacency in G A G given by {v ± , v 2 ) ~ (u x , w 2 ) i f (and only if) v. ~ u, in G and «" ~ w in (?"…”