‎    The Narumi-Katayama index is the first topological index defined by the product of some graph theoretical quantities. Let G be a simple graph. Narumi-Katayama index of G is defined as the product of the degrees of the vertices of G. In this paper, we define the Narumi-Katayama polynomial of G. Next, we investigate some properties of this polynomial for graphs and then, we obtain ...

To compare several promising product designs manufacturers must measure their performance under multiple environmental conditions In many applications a product design is considered to be seriously awed if its performance is poor for any level of the environmental factor For example if a particular automobile battery design does not function well under temperature extremes then a manufacturer m...

We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered by two vertex-disjoint monochromatic paths of distinct colours, under the necessary assumption that the largest partition class of G contains at most n/2 vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, ...

We present an algorithm that supports operations for modifying a split graph by adding edges or vertices and deleting edges, such that after eachmodification the graph is repaired to become a split graph in a minimal way. In particular, if the graph is not split after the modification, the algorithm computes a minimal, or if desired even a minimum, split completion or deletion of themodified gr...

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexi...

This paper shows how to embed complete binary trees in products of complete binary trees products of shu e exchange graphs and products of de Bruijn graphs The main emphasis of the embedding methods presented here is how to emulate arbitrarily large complete bi nary trees in these product graphs with low slowdown For the embedding methods presented here the size of the host graph can be xed to ...

A simple graph-product type construction shows that for all natural numbers r ≥ q, there exists an edge-coloring of the complete graph on 2 vertices using r colors where the graph consisting of the union of arbitrary q color classes has chromatic number 2. We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the ...

Product codes are widely used in data-storage, optical and wireless applications. Their analytical performance evaluation usually relies on the truncated union bound, which provides a low error rate approximation based on the minimum distance term only. In fact, the complete weight enumerator of most product codes remains unknown. In this paper, concatenated representations are introduced and a...