The first and second Zagreb indices of some graph operations

This survey outlines results on graphs extremal with respect to distance–based indices, with emphasis on the Wiener index, hyper–Wiener index, Harary index, Wiener polarity index, reciprocal complementary Wiener index, and terminal Wiener index.

A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices

The Zagreb coindices of graph operations

On the topological index

The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.

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Mathematical aspects of Wiener index

Vertex and edge PI indices of Cartesian product graphs

We study distance-based graph invariants, such as the Wiener index, the Szeged index, and variants of these two. Relations between the various indices for trees are provided as well as formulas for line graphs and product graphs. This allows us, for instance, to establish formulas for the edge Wiener index of Hamming graphs, C"4-nanotubes and C"4-nanotori. We also determine minimum and maximum of certain indices over the set of all graphs with a given number of vertices or edges. Finally, we study the order of magnitude of the edge Wiener and edge Szeged index, responding negatively to a conjecture that is related to the maximization of the edge Szeged index.

Some new results on distance-based graph invariants

Let G be a graph. The distance d(u,v) between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as WW(G)=12W(G)[email protected]?"{"u","v"}"@?"V"("G")d(u,v)^2. In this paper the hyper-Wiener indices of the Cartesian product, composition, join and disjunction of graphs are computed. We apply some of our results to compute the hyper-Wiener index of C"4 nanotubes, C"4 nanotori and q-multi-walled polyhex nanotori.

The hyper-Wiener index of graph operations

The edge Szeged index of a molecular graph G is defined as the sum of products m u (e|G)m v (e| G) over all edges e = uv of G, where m u (e|G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and where m v (e| G) is defined analogously. The aim of this paper is to compute the edge Szeged index of the Cartesian product of graphs. As a consequence of our result, the edge Szeged index of Hamming graphs and C 4 -nanotubes are computed.

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M. H. Khalifeh

Papers

The first and second Zagreb indices of some graph operations

Vertex and edge PI indices of Cartesian product graphs

Some new results on distance-based graph invariants

The hyper-Wiener index of graph operations

The Edge Szeged Index of Product Graphs