Invariant, vertex

While the rows and columns of A obviously depend on a particular choice of vertex labels, the generic structural j)roperties of G must remain invariant under a permutation of rows and columns. Much of this structural information can in fact be extracted from the spectrum of G the spectrum of a graph G,... 

To answer the question one has to examine carefully the permutations which correspond to the 24 rotations of the octahedron. We partition these permutations into cycles and assign to each cycle of a certain order k the symbol f. assign to a cycle of order 1 (vertex which is invariant under rotation), f to a cycle of order two (transposition), /g to a cycle of order three, etc. A permutation which is decomposed into the product of cycles with no common elements is represented by the product of the symbols /. associated with the corresponding cycles. Thus the rotations of the octahedron are described by the following products ... 

Balaban, A. T., and M. V. Diudea, Real Number Vertex Invariants Regressive Distance Sums and Related Topological Indices. J. Chem. Inf. Comput. Sci., 1993 33 421-428. 

Diudea, M. V., O. Minailiuc, and A. T. Balaban, Regressive Vertex Degree (New Graph Invariants) and Derived Topological Indices. J. Comput. Chem., 1991 12, 527-535. 

The results of the analysis for a system of three isomers for various E are represented in Fig. 9(a)-(b). Here, a convex envelope for the finite multitude (106) is vertically hatched and its union with the multitude G(N) e( ) is horizontally hatched. The whole of the hatched multitude is co-invariant and the unhatched region is just V(E). This example of only four multitudes makes it possible to construct the "unattainability regions that would not be a union of those for submultitudes. Three multitudes each contain one vertex and a fourth [Fig. 9(d)] includes two vertices, corresponding to the cases when the entire mass is concentrated either in Aj or in A2. 

Ivanciuc, O., Balaban, T.S., and Balaban, A.T., Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem., 12 309-318, 1993. 

Balaban S, Filip PA, Ivanciuc O, Computer generation of acyclic graphs based on local vertex invariants and topological indices derived canonical labelling and coding of trees and alkanes, J. Math. Chem., 11 79-105, 1992. 

The next step of invariant construction is the introduction of two-state spins in all vertices. The states of each spin correspond to two possible ways of vertex splitting... 

In search of invariants. Are there possibly any other characteristics of a graph (or its topological matrix) that are independent of the vertex enumeration mode Yes, such invariant characteristics do exist. However, they can be obtained only after certain refinements of the theory. 

Next seven invariants are based on the vertex distance D v) = d u, v) and the graph distance D YLv v... 

The regular orbit of a point symmetry group is the set of positions for which the only operation that leaves each vertex label invariant is the identity all other operations permute... 

Moreover, the adjacency matrix can be transformed into a decimal adjacency vector a of A elements each being a local vertex invariant obtained by the following expression [Schultz and Schultz, 1991] ... 

To obtain spatial autocorrelation molecular descriptors, function /(x,) is any physico-chemical property calculated for each atom of the molecule, such as atomic mass, polarizability, etc., and - local vertex invariants such as - vertex degree. Therefore, the molecule atoms represent the set of discrete points in space and the atomic property the function evaluated at those points. 

These are global molecular descriptors derived from an H-depleted molecular graph where each vertex is weighted by a local vertex invariant called Atom-in-Structure Invariant Index (ASII) defined as [Bangov, 1988] ... 

Some molecular descriptors and local vertex invariants proposed as a generalization or modification of the original connectivity indices are reported below. 

The maximum path sum of the / th vertex, denoted by MPVS, is a local vertex invariant defined as the sum of the length of the longest paths between vertex v, and any other vertex in the molecular graph, i.e. 

The maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

All these quantities are -> local vertex invariants. High values of the vertex distance sum o are observed for -> terminal vertices while low values for -> central vertices. Moreover, among the terminal vertices, the vertex distance degrees are small if the vertex is close to a branching site and larger if the terminal vertex is far away. 

The minimum value of the vertex distance degrees is another molecular invariant called the unipolarity ... 

The reciprocal distance sum RDSi of the i th vertex is a local invariant defined as the sum of the reciprocal distance matrix elements in the i th row ... 

Bond multiplicity is taken into account by augmenting the edge distance matrix with a supplementary column and row where the elements are conventional bond orders, therefore obtaining an edge distance matrix for multigraphs [Bonchev, 1983]. All the local vertex invariants and molecular descriptors defined above can also be calculated on this matrix. 

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