Complexity of a Graph

One of the simplest ways of modeling a variety of physical structures, from molecules to rigid three-dimensional structures, is the mathematical graph. 

Perhaps the simplest approach to defining a complexity of graphs is to apply Shannon s information measure to its vertices [dres74]. Let Vy = Vi 1 

I Vy 1= h V) depends on the partition. Let V) be the number of nodes in the set Vi- Then the complexity X G) of the graph G is given by minimizing Shannon s information over all possible partitions of this vertex set  

An automorphism of a graph G is an isomorphism of G onto itself. It is easy to 

The topological complexity, Ct(G), of a graph G is then defined by [mowshbSa]  

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B G) is often used to define the complexity of a graph G, (G), equal to the number of spanning trees of G. It turns out that... 

Graph Complexity Static Captures intuitive feel of the complexity of a graph Not a-priori clear how to extend to more general objects... 

ICr Mean information content or complexity of a graph based on the rth (r — 0-6) order... 

Several of the works discussed above include graph-theoretic calculations of, for example, the complexity of a graph. Unfortunately it is not clear in many cases what the implications are for the reaction network for differences in the complexity of various associated graphs, particularly when the differences are small. In some cases, the results seem counterintuitive in that the more complex graph is constructed from the physically more important reaction process. More study is needed of these issues. 

Mowshowitz, A. (1968a). Entropy and the Complexity of Graphs. I. An Index of the Relative Complexity of a Graph. Bull.Math.Biophys.,30,175-204. 

Mowshowitz, A. (1968a) Entropy and the complexity of graphs. I. An index ofthe relative complexity of a graph. Bull. Math. Biophys., 30, 175-204. 

Mean information content or complexity of a graph based on the r (r = 0-6) order neighborhood of vertices in a hydrogen-filled graph Structural information content for r" (r = 0-6) order neighborhood of vertices in a hydrogen-filled graph... 

CIC, represents the difference between maximum possible complexity of a graph (where each vertex belongs to a separate equivalence class) and the realized topological information of a chemical species as defined by IC,. Figure 6 provides an example of the first order (r = 1) calculations of IC, SIC, and CIC. 

The neighborhood complex of a graph is an example in which this dual description is not useful. On the other hand, the matroid complex is an example in which both descriptions are frequently used, depending on the circumstances. The dual presentation in this case uses the term circuits, which play the role of the minimal nonsimplices, and goes as follows. 

Special Families of Horn Complexes 18.2.1 Coloring Complexes of a Graph... 

We conclude this section by mentioning two older definitions of complexity, each of which also depends on both the size and vertex structure of a graph G (1) the number of spanning trees in Gj and (2) the average number of independent paths between vertices in G. 

The classical procedure is to mix P and X and dilute to constant volume so that the total concentration [P] + [X] is constant. For example, 2.50 mM solutions of P and X could be mixed as shown in Table 19-1 to give various X P ratios, but constant total concentration. The absorbance of each solution is measured, typically at max for the complex, and a graph is made showing corrected absorbance (defined in Equation 19-21) versus mole fraction of X. Maximum absorbance is reached at the composition corresponding to the stoichiometry of the predominant complex. 

A closed map, cell-complex of a polyhedron. It is a 5Rq, 4/ 2 plane graph (see Chapter 9)... 

The validation results shown in this specific example might lead one to make a generalized rule that the optimal complexity of a model corresponds to the level at which the RMSEP is at a minimum. However, it is not always the case that RMSEP-versus-complexity graph shows such a distinct minimum, and therefore such a generalized rule can result in overfit models. Alternatively, it might be more appropriate to choose the model complexity at which an increase in complexity does not significantly decrease the prediction error (RMSEP). This choice can be based on rough visual inspection of the prediction error-versus-complexity plot, or from statistical tools such as the/-test.50,51... 

In his classic paper on electric networks, G. Kirchhoff[38] (1847) implicitly established the celebrated Matrix-Tree-Theorem which, in modern terminology, expresses the complexity (i.e., the number of spanning trees) of any finite graph G as the determinant of a matrix which can easily be obtained from the adjacency matrix of G. Simple proofs were given by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte [39] (1940), H. Trent [40] (1954), and H. Hutschenreuther [41] (1967) (for relations between the complexity and the spectrum of a graph see Ref. [36] pp 38, 39, 49, 50). 

A simple measure of structural complexity of a molecule based on its graph representation G compared with the parent -> complete graph K(Q, i.e. the complete graph with the same number of vertices. It is defined as ... 

Some - shape descriptors and -> centric indices contain information about symmetry, while -> WHIM symmetry, - Bertz complexity index, - indices of neighbourhood symmetry, and - symmetry number obtained by the automorphism group of a graph are explicitly related to the symmetry. 

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