Connected labeled graphs

As the objects we are interested in this chapter are connected, let Ck be the number of connected labeled graphs of k vertices. kC rooted connected labeled graphs exist because k ways exist of choosing a root. The number of rooted, labeled graphs of n vertices in which the root is in a connected component containing k vertices is feQ ( This expression, summed from... 

Finally we have to decompose the product graphs into connected components and to eliminate the isomorphic duplicates. For this reason we introduce the set of connected labeled graphs for an arbitrary set of molecular graphs... 

The constitution is customarily represented by constitutional formulas, i.e. by labelled graphs whose nodes are the atoms, and whose connecting lines are the bonds. Nodes are labelled by chemical element symbols and can carry further symbols for electrons and electrical charges. 

To simplify comparisons between reactions, we sought to describe entire reactions as a single labeled graph. Just as cause and effect can be considered either as two separate events or as a unified process, changing with time, so a reaction can be perceived as two structures, as shown above, or as a single assembly of nuclei connected by bonds which change with time. The aldol-type reaction just illustrated can be rewritten as follows ... 

Pi(l) = the sum of all distinct connected simple graphs consisting of one white Zj-circle labeled 1, some or no black Zi-circles, and /-bonds (2.1.21)... 

Such SRSG s are labelled graphs which can be easily recorded using connectivity tables based on some arbitrary (or canonical) numbering of their atoms (vertices). Such numbers are represented in the above examples by encircled Greek letters. Allowing one b)die for an atom symbol, six bits for an SRSG atom number and four bits for an... 

If y 9, then the orbit S ((e, y)) e S lvt of the labeled molecular graph (e, y) is called a connected unlabeled molecular graph. Equivalence classes of connected labeled molecular graphs can be identified with constitutional formulas of chemical compounds. Exceptions are exotic compounds such as catenanes and rotaxanes, whose structural formulas correspond to disconnected graphs. These are not dealt with in this book. The notation is... 

Moreover, we assume a transversal T of this set of labeled connected molecular graphs, together with a function... 

The vertex-adjacency matrix or binary matrix, denoted by A, of a vertex-labeled connected simple graph G with Vvertices is a square VxVmatrix, which is determined by the adjacencies of vertices in G (Harary, 1971) ... 

Lukovits (2000, 2002, 2004) and Lukovits and Gutman (2002) offered an approach by which the vertex-adjacency matrix of an acyclic structure can be replaced by a single number, called the compressed (vertex-) adjacency matrix code, denoted by CAM. Here we present, besides the CAM code, the N-tuple code of trees that induces the unique labeling of trees (Aringhieri et al 1999). A graph is acyclic if it does not contain cycles. A tree is a connected acyclic graph. 

Connection tables record the constitution of the molecule using a node and edge labeled graph. The nodes of the graph are arbitrarily numbered and can be stored as ... 

Figure 6-1. Different forms of representation of a chemical graph a) labeled (numbered) graph b) adjacency matrix c) connectivity table, type I d) connectivity table, type II f) line notations g) structural index.

A series of four papers by G. W. Ford and others [ForG56,56a,56b, 57] amplified this work by using Polya s Theorem to enumerate a variety of graphs on both labelled and unlabelled vertices. These included connected graphs, stars (blocks) of given homeomorphic type, and star trees. In addition many asymptotic results were derived. The enumeration of series-parallel graphs followed in 1956 [CarL56], and in that and subsequent years Harary produced... 

Make a graph with an energy scale extending on the ordinate from zero to 3000 kcal/mole and with the abscissa marked at equal intervals with the labels Na, Mg, and Al. Now plot and connect with a solid line the first ionization energies, Ei, of these three elements (see Table 20-IV). Hot Et and connect with a dashed line, E with a dotted line, and Et with a solid line. Draw a... 

Using graph paper and your data from Part A, construct a temperature (y-axis) versus time (x-axis) graph for the cooling of the naphthalene. Do not connect the data points. Label this graph Cooling Curve of Pure Naphthalene. ... 

The scheme in P in Example IV-2 is not tree-like. (Here we let A,B,C,D,E stand for arbitrary assignment statements or sets of assignment statements, in order to exhibit the structure of the example uncluttered by extraneous formulae.) The direct connection from the node labelled T(x ) following the node labelled D up to the node labelled B is anomalous since the path START A T(x ) D T(x ) does not contain B and the path START A T(x- ) B does not contain the node labelled T(x- ) following D. Similarly the two direct connections at the bottom of the diagram, from test T(x2) to test T(x ) and from test T(x ) to E are anomalous. The other direct connections in the graph are not anomalous. For example, the direct connection from test T(x2> up to the node labelled A is not anomalous since A is an ancestor of every node in the graph except START. 

Scheme P is a graph hononorphic image of the scheme P in Example IV-3, which is tree-like. Scheme P is formed from P by duplicating nodes when anomalous connections occur. Thus the direct connection from the node labelled T(x ) which follows D, to B is anonalous in P. In P, this connection is removed and a direct connection to a copy of B is substituted. The copy of B leads into a copy of the subscheme reachable from B until A occurs. We do not need to duplicate A since any connection to A must be acceptable hence the diagram loops lack to A. Similarly, the anomalous direct connection from the node T(x ) which follows C to E is anonalous and is replaced by a connection to a copy of E and the node following E, TCxg) the connections from this instance of I(x2) are now legal and so no duplicates are needed.

Graph molar heats of fusion versus atomic p number. Connect the points with straight f = lines and label the curve. Do the same for = = =l molar heats of vaporization. 

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