Graphs connectedness

The total adjacency index Ay is a measure of the graph connectedness and is calculated as the sum of all the entries of the adjacency matrix of a molecular graph, vhich is tv ice the number B of graph edges [Harary, 1969a Rouvray, 1983] ... 

WriE72 Wright, E. M. The probability of connectedness of an unlabelled graph can be less for more edges. Proc. Am. Math. Soc. 35 (1972) 21-25. 

We required 2-connectedness and that any two holes do not share a vertex. If we remove those two hypothesis, then many other graphs do appear. 

The most well-known complexity indices are listed below. Other molecular descriptors which give information about molecular complexity are the indices of neighbourhood symmetry and the - total adjacency index. The latter was proposed by Bonchev and Polansky [Bonchev and Polansky, 1987] as a simple measure of topological complexity, being a measure of the degree of connectedness of molecular graph. 

The complexity index can certainly be calculated for graphs with pendant vertices. In this case one takes into account the fact that the pendant vertices do not change the mechanism complexity because the basic graph topology (number of cycles, number of vertices in the graph s cyclic part, the cycle connectedness, etc.) is left unchanged. The presence of pendant vertices becomes of importance in the case of nonstationary processes. For stationary (or... 

As these ideas were developed, it appeared appropriate to give a name to this approach. It was decided to use a name that brings together graph notions, such as adjacency and connectedness, with the central focus of this method, the chemical and molecular nature of structure. The name molecular connectivity was selected. ... 

The first-order chi indexes contain more structure information than do the zero-order indexes. The immediate bonding environment of each atom is encoded by virtue of the edge weight. Further, the number of terms in the sum, P, is dependent on the graph type, especially on the number of cycles or rings. The x index encodes both the atom identities as well as the connectedness in the molecular skeleton. 

Due to the connectedness of, this is a connected graph. It is possible to make a partition... 

A very useful representation of the Markov Chain ttansition matrix II is by use of a weighted directed graph in which each node represents a state of the Markov Chain and the weights represent transition probabilities. Then one may understand irreducibility as expressing the connectedness of this graph (see Fig. 6.6). 

The 3-connectedness of fuUerene graphs results in an obvious upper bound D(G) < n/3. It is intuitively clear that linear upper bounds are actually achieved in long tubular fullerenes. It is also clear that the constant must depend on the circumference of the tubular part. Since the tubular fullerenes exist for all values of n for which there is a fuUerene graph on n vertices, it follows that no sublinear upper bound is possible for general fuUerenes. The above reasoning was formalized in a recent paper by Andova et al. (2010). Their main result is that the diameter of almost aU fuUerenes on n vertices is at most n/6 -H 2.5. 

Connectedness is an equivalence relation on the set n of nodes of y. It decomposes n into pairwise disjoint sets of connected nodes (together with the edges in between), tbe connected components of the graph. The set of connected components of y is denoted by... 

Relaxing connectedness Disconnected molecular graphs may or may not be allowed. The implementation applies that the degree partition of a simple graph is the degree partition of a connected simple graph if and only if B(y) > n - 1 (see the items following Remark 2.35). 

Erdos, P. and Renyi, A. (1960) On the strength of connectedness of a random graph. Acta Math. Hung., 12, 261-267-... 

So as to verify the connectedness of a graph, it is sufficient to select one (arbitrary) node n if for any other node n n) there exists a path of endpoints n and n then the graph is connected. 

Molecular graphs, complete or incomplete, can be conveniently represented by connectivity matrices as can be seen in the examples in Fig. 3.1 and in Refs. [1-9]. A complete graph where connectedness is indicated by 1 and disjointedness by 0 will have a non-zero entry for every non-diagonal element of the matrix while an incomplete graph has finite entries only for connected vertices and zero elsewhere in the matrix (Fig. 3.1). 

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