This paper introduces two kinds of graph polynomials, clique polynomial and independent set polynomial.The paper focuses on expansions of these polynomials. Some open problems are mentioned.

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The concept of a line graph is generalized to that of a path graph. The path graph f,(G) of a graph G is obtained by representing the paths Pk in G by vertices and joining two vertices whenever the corresponding paths f k in G form a path f k + , or a cycle C,. f,-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian /?,-graphs are characterized.

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A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. The discovery of these integral complete r-partite graphs is a new contribution to the search of such integral graphs. Finally, we propose several basic open problems for further study.