Basic Chemical Graph Theory

Abstract

Graph Theory applied in Chemistry is called Chemical Graph Theory. This inter-disciplinary science takes problems (like isomer enumeration, structure elucidation, etc.) from Chemistry and solve them by Mathematics (using tools from Graph Theory, Set Theory or Combinatorics), thus influencing both Chemistry and Mathematics. Partitioning of a molecular property and reconstructing it from fragmental contributions is one of the main tasks of this theory. For further discussion, some basic definitions in Graph Theory are needed. 1.1 Basic Definitions in Graphs A graph G(V, E) is a pair of two sets, V and E, V ¼ V(G) being a finite nonempty set and E ¼ E(G) a binary relation defined on V (Harary 1969). A graph can be visualized by representing the elements of V by points/vertices and joining pairs of vertices (i, j) by an edge/bond if and only if (i, j) 2 E(G). The number of vertices in G equals the cardinality n ¼ jV(G)j of this set. The term graph was introduced by Sylvester (1874). There is a variety of graphs, some of them being mentioned below. A path graph is a non-branched chain. A tree is a branched structure. A star is a set of vertices joined in a common vertex. A cycle is a chain which starts and ends in one and the same vertex (Fig. 1.1). A complete graph K n is the graph of with any two vertices are adjacent. The number of edges in such a graph is n(n À 1)/2. Fig. 1.2 illustrates the complete graphs with n ¼ 1-5. In a bipartite graph, the vertex set V can be partitioned in two disjoint subsets: V 1 [ V 2 ¼ V(G); V 1 \ V 2 ¼ ∅ such that any edge (i, j) 2 E(G) joins V 1 with V 2 (Harary 1969; Trinajstic´1983Trinajstic´Trinajstic´1983; Diudea 2010). A graph is bipartite if and only if all its cycles are even (Ionescu 1973). If any vertex i 2 V 1 is adjacent to any vertex j 2 V 2 then G is a complete bipartite graph, K m,n , with m ¼ |V 1 | and n ¼ |V 2 | and the