Abstract:
Expressions for the path polynomials (see Farrell [1]) of chains and circuits are derived. These polynomials are then used to deduce results about node disjoint path decompositions of chains and circuits. Some results are also given for decompositions in which specific paths must be used.

Abstract:
The simple tree polynomials of the basic graphs with cyclomatic number 3 are derived. From these results, explicit formulae for the number of decompositions of the graphs into forests with specified cardinalities are extracted. Explicit expressions are also given for the number of spanning forests and spanning trees in the graphs. These results complement the results given in [1].

Abstract:
The clique polynomial of a graph is defined. An explicit formula is then derived for the clique polynomial of the complete graph. A fundamental theorem and a reduction process is then given for clique polynomials. Basic properties of the polynomial are also given. It is shown that the number theoretic functions defined by Menon are related to clique polynomials. This establishes a connection between the clique polynomial and decompositions of finite sets, symmetric groups and analysis.

Abstract:
A special type of family graphs (F-graphs, for brevity) are introduced. These are cactus-type graphs which form infinite families under an attachment operation. Some of the characterizing properties of F-graphs are discussed. Also, it is shown that, together with the attachment operation, these families form an infinite, commutative semigroup with unit element. Finally, it is shown that F-graphs are graph-theoretical representations of natural numbers.

Abstract:
Explicit recurrences are derived for the matching polynomials of the basic types of hexagonal cacti, the linear cactus and the star cactus and also for an associated graph, called the hexagonal crown. Tables of the polynomials are given for each type of graph. Explicit formulae are then obtained for the number of defect-d matchings in the graphs, for various values of d. In particular, formulae are derived for the number of perfect matchings in all three types of graphs. Finally, results are given for the total number of matchings in the graphs.

Abstract:
Explicit formulae, in terms of sugraphs of the graph, are given for the first six coefficients of the simple path polynomial of a graph. From these, explicit formulae are deduced for the number of hamiltonian paths in graphs with up to six nodes. Also simplified expressions are given for the number of ways of covering the nodes of some families of graphs with k paths, for certain values of k.

Abstract:
Let G be a graph. With every path ± of G let us associate a weight w ± With every spanning subgraph C of G consisting of paths ±1, ±2, ￠ € |, ±k, let us associate the weight w(C)= ￠ i=1kw ±i. The path polynomial of G is ￠ ‘w(C), where the summation is taken over all the spanning subgraphs of G whose components are paths. Some basic properties of these polynomials are given. The polynomials are then used to obtain results about the minimum number of node disjoint path coverings in graphs.

Abstract:
Results are given from which expressions for the coefficients of the simple circuit polynomial of a graph can be obtained in terms of subgraphs of the graph. From these are deduced parallel results for the coefficients of the characteristic polynomial of a graph. Some specific results are presented on the parities of the coefficients of characteristic polynomials. A characterization is then determined for graphs in which the number of sets of independent edges is always even. This leads to an interesting link between matching polynomials and characteristic polynomials. Finally explicit formulae are derived for the number of ways of covering two well known families of graphs with node disjoint circuits, and for the first few coefficients of their characteristic polynomials.

Abstract:
The Subgraph polynomial fo a graph pair (G,H), where H ￠ …G, is defined. By assigning particular weights to the variables, it is shown that this polynomial reduces to the dichromatic polynomial of G. This idea of a graph pair leads to a dual generalization of the dichromatic polynomial.