Graph connected

A graph is connected if at least one edge is between all the nodes. Thus, from any given node in a connected graph, all the other nodes can be reached. 

Puler path A connected graph can be traversed in one path (W hich ends at the node where it began) ifall nodes have an even degree (sec the Konigsberg bridge problem. Section 2.4.1). 

Fig. 12.1 Graphs contain nodes connected by edges. A completely connected graph (right) has an edge between all pairs of nodes.

Table 5.2 Number of topologically distinct connected graphs ) ), number of cyclic equivalence classes Q, maximal numbers of possible cycle sets Cot and Ct for OT and T rules, respectively, and the maximal number of possible distinct topologies of the state transition graph, calculated for graphs with size fV=5,6,..., 12 in T 2. ...

If we remove from such graphs an endpoint including the edge belonging to the endpoint and if we keep repeating this process as often as possible, we end up with a ring that is, a connected graph of m... 

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... 

If we progressively delete vertices of degree 1 from a connected graph C, until no more such vertices remain, we shall obtain a connected graph F which we can call the "frame" of C. The graph C is then seen to consist of the graph F at each vertex of which a rooted tree has been attached, so that the root of the tree is identified with the vertex of F. Figure 8 shows a typical example. 

HarF55 Harary, F. The number of linear, directed, rooted and connected graphs. Trans. Amer. Math. Soc. 78 (1955) 445-463. 

A finite connected graph is said to be unicursal if all its edges can be traversed (or traced) continuously. An Euler graph is one in which... 

In a simple (nonweighted) connected graph, the graph distance dy between a pair of vertices V and Vj is equal to the length of the shortest path cormecting the two vertices, i.e. the number of edges on the shortest path. The distance between two adjacent vertices is 1. The distance matrix D(G) of a simple graph G with N vertices is the square NxN symmetric matrix in which [D],j=cl,j [9, 10]. 

In the above MILP-optimization problem, Euler s theorem for the generation of stable and feasible molecular structures (fully connected graphs) needs to be added as a condition in order to ensure the generation of chemically feasible molecules. This condition is mathematically formulated as,... 

The inspection of the results obtained for all connected graphs of up to six vertices motivates the conclusions collected in Table 2. 

We remind that an automorphism of a simple graph is a permutation of the vertices preserving adjacencies between vertices. For plane graphs, we require also that faces are sent to faces but for 3-connected graphs this condition is redundant. Recall that Aut(G) denotes the group of automorphisms of G. 

Given an (r, -polycycle P, its major skeleton Maj(P) is the plane graph formed by its elementary components with two components being adjacent if they share an open edge. A tree is a connected graph with no cycles. 

It is easy to see that cycles of length 2 and 3, not enclosing a vertex, do not exist, since, otherwise, we would get only a 1-connected graph, while ( 5,6, 3)-spheres are 3-connected (see Theorem 2.0.2). 

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