design of crystal-like lattices can be achieved by using some net operations. hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. the networks herein presented are related to the dyck graph and described in terms of omega polynomial and piv polynomials.

This paper designs a set of graph operations and proves that starting from Gd , by repeatedly applying these operations, one can construct all graphs G with χc(G) ≥ k/d (for k/d ≥ 3). This can be viewed as an analogue of Hajós’ Theorem for the circular chromatic number.

This paper designs a set of graph operations and proves that starting from Gd , by repeatedly applying these operations, one can construct all graphs G with χc(G) ≥ k/d (for k/d ≥ 3). This can be viewed as an analogue of Hajós’ Theorem for the circular chromatic number.

Design of crystal-like lattices can be achieved by using some net operations. Hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. The networks herein presented are related to the Dyck graph and described in terms of Omega polynomial and PIv polynomials.

The chromatic capacity χcap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f :N → N such that χcap(G) ≥ f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show tha...

Graph complexity measures like tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is de ned in ...

Let G = (V, E) be a graph. For e = uv ∈ E(G), nu(e) is the number of vertices of G lying closer to u than to v and nv(e) is the number of vertices of G lying closer to v than u. The GA2 index of G is defined as ∑ uv∈E(G) 2 √ nu(e)nv(e) nu(e)+nv(e) . We explore here some mathematical properties and present explicit formulas for this new index under several graph operations.