Abstract
The degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. The restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. This article provides some exact values for the said problem and also gives some bounds for the optimal graphs.
Highlights
All graphs discussed in paper are simple, finite, and undirected. e valency of a node in the graph G is the number of edges connected with that node in G. e maximal valency of the graph G is indicated by Δ(G)
By the degree of a node, it is meant to have the number of connections attached to that node; on the contrary, the diameter shows the largest number of links that must be required to transmit a message between any two nodes. e natural question that arises in this case is
We have considered the restricted version of Degree Diameter Problem which states that given connected undirected host graph G, an upper bound D for the maximum degree, and an upper bound Δ for the diameter find the largest connected subgraph of maximum degree ≤ Δ and diameter ≤ D. is problem becomes of particular interest when we consider the host graph as the network
Summary
All graphs discussed in paper are simple, finite, and undirected. e valency (or degree) of a node (or vertex) in the graph G is the number of edges connected with that node in G. e maximal valency of the graph G is indicated by Δ(G). E natural question that arises in this case is “What is the largest number of nodes in a network with a limited degree and diameter?” If we design the network so that there is no directed edge, this leads to the Degree/Diameter Problem. In the Degree Diameter Problem, only restriction that is imposed on the edges is the maximum degree, so there is considerable freedom in placing edges so as to avoid violating the diameter constraint In this way, the resulting graph may be thought of as being embedded in the complete graph. E grid graph network Pm□Pn has mn nodes and 2mn − m − n edges. (i) e edge connected to the hub and left-end vertex of upper horizontal edge in W5 is called upper-left hub edge (ii) e edge connected to the hub and right-end vertex of upper horizontal edge in W5 is called upper-right hub edge (iii) e edge connected to the hub and left-end vertex of lower horizontal edge in W5 is called lower-left hub edge (iv) e edge connected to the hub and right-end vertex of lower horizontal edge in W5 is called lower-right hub edge
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