The degree-based entropy Id(G) of a graph G on m>0 edges is obtained from the well-known Shannon −∑i=1np(xi)logp(xi) in information theory by replacing probabilities p(xi) fractions dG(vi)2m, where {v1,v2,…,vn} vertex set G, and dG(vi) degree vi. We continue earlier work Id(G). Our main results deal with effect number operations value also illustrate relevance these applying some to prove extre...

We prove that counting copies of any graph F in another graph G can be achieved using basic matrix operations on the adjacency matrix of G. Moreover, the resulting algorithm is competitive for medium-sized F : our algorithm recovers the best known complexity for rooted 6-clique counting and improves on the best known for 9-cycle counting. Underpinning our proofs is the new result that, for a ge...

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

We show that an operation on graphs, and more generally, on relational structures that has an inverse definable by a monadic secondorder transduction preserves the family of recognizable sets.

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graphG, say S, is a set of vertices inGwhose closed neighborhood is removed from G. The survival subgraph is denoted by G/S. The neighbor rupture degree of G, Nr(G), is defined to...

In this paper, we outline our work on developing a disk-based infrastructure for efficient visualization and graph exploration operations over very large graphs. The proposed platform, called graphVizdb, is based on a novel technique for indexing and storing the graph. Particularly, the graph layout is indexed with a spatial data structure, i.e., an R-tree, and stored in a database. In runtime,...

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)+12@?"{"u","v"}"@?"V"("G")d (u,v)^2. In this paper the hyper-Wiener indices of the Cartesian product, composition,...

The connective eccentric index of a graph is a topological index involving degrees and eccentricities of vertices of the graph. In this paper, we have studied the connective eccentric index for double graph and double cover. Also we give the connective eccentric index for some graph operations such as joins, symmetric difference, disjunction and splice of graphs. MSC (2010): Primary: 05C35; Sec...

A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. A bipolar fuzzy graph structure is a generalization of a bipolar fuzzy graph. In this paper, we present several different types of operations, including composition, Cartesian product, strong product, cross product, and lexicographic product on ...