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.
A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterizing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis.
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… (More)
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… (More)
The maximum clique problem is one of the NP-complete problems. There are graphs for which a reduction technique exists that transforms the problem for these graphs into one for graphs with specific properties in polynomial time. The resulting graphs do not grow exponentially in order and number. Graphs that allow such a reduction technique are called soft.… (More)