A very brief survey of the main results concerning the cell–growth problem and its variations is given. The name stems from an analogy with an animal which, starting from a single cell of some specified basic polygonal shape, grows step by step in the plane by adding at each step a cell of the same shape to its periphery. The fundamental combinatorial problem concerning these animals is ”How ma...

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type in...

By extending Heider’s and Cartwright-Harary’s theory of balance in deterministic social structures, we study the problem of balance in social structures where relations between individuals are random. An appropriate model for representing such structures is the so called random signed graphs Gn,p,q defined as follows. Given a set of n vertices and fixed numbers p and q, 0 < p+ q < 1, then, betw...

A graph is called integral if all its eigenvalues are integers. The quest for integral graphs was initiated by F. Harary and A. J. Schwenk [9]. All such cubic graphs were obtained by D. Cvetković and F. C. Bussemaker [2, 1], and independently by A. J. Schwenk [10]. There are exactly thirteen cubic integral graphs. In fact, D. Cvetković [2] proved that the set of regular integral graphs of a fix...

If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M...

A tournament T,, consists of a set of n vertices and a single directed edge joining every pair of distinct vertices. We denote the vertices of T’. by {l, . . . , n). A permutation aI, . . . , a, of the vertices is a generalized path. The type af the path is characterized by the sequence a,_1 = e3 l l l E~__~ where si = i-1 if e -+ ai+1 and 6 = -1 if a, -+ ai tl. We also say that mn_l is realize...

In [8], a high availability framework based on Harary graph as network topology has been proposed for stateful failover. Framework proposed therein exhibits an interesting property that an uniform load can be given to each non-faulty node while maintaining fault tolerance. A challenging problem in this context, which has not been addressed in [8] is to be able to come up with a distributed algo...

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric r-dissections of polygons and planar S-dissections are obtained.