Then x and y are said to be adjacent , and the edge x , y is incident with x and y. If x , y is not an edge, then the vertices x and y are said to be nonadjacent. G is a finite graph if V G is finite. The length of a chain is the number of edges in it. Two isomorphic graphs count as the same unlabelled graph. A graph is said to be a tree if it contains no cycle—for example, the graph G 3 of Figure 3.

A rooted tree has one point, its root, distinguished from others. Polya in showed in his memoir already referred to that the generating function for rooted trees satisfies a functional equation. If there is a class C of graphs each of which possesses a certain set of properties P , then the set of properties P is said to characterize the class C , provided every graph G possessing the properties P belongs to the class C.

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Sometimes it happens that there are some exceptional graphs that possess the properties P. Many such characterizations are known.

## Combinatorics/Graph & Ramsey Theory

Here is presented a typical example. A complete graph K m is a graph with m vertices, any two of which are adjacent. The line graph H of a graph G is a graph the vertices of which correspond to the edges of G , any two vertices of H being adjacent if and only if the corresponding edges of G are incident with the same vertex of G. A graph G is said to be regular of degree n 1 if each vertex is adjacent to exactly n 1 other vertices. A strongly regular graph and a two-class association are isomorphic concepts.

The treatments of the scheme correspond to the vertices of the graph, two treatments being either first associates or second associates according as the corresponding vertices are either adjacent or nonadjacent. Pay Enrollment Deposit. Financial Aid. Estimate Your Cost. More Programs. College Courses in High School.

### Topics in Combinatorics and Graph Theory - 0366.4771.01

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## Combinatorics, Probability and Algorithms @ Bham

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