The vertex PI index PI(G) = ∑ xy∈E(G)[nxy(x) + nxy(y)] is a distance-based molecular structure descriptor, where nxy(x) denotes the number of vertices which are closer to the vertex x than to the vertex y and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common ver...

In recent work [1] we have found identical behaviour for various spin models on " thin " random graphs-Feynman diagrams-and the corresponding Bethe lattices. In this paper we observe that the ratios of the saddle point equations in the random graph approach are identical to the fixed point(s) of the recursion relations which are used to solve the models on the Bethe lattice. The loops in the ra...

Chromosome numbers and meiotic behavior are reported for the climbing cacti species Hylocereus undatus, Hylocereus polyrhizus, and Selenicereus megalanthus. The Hylocereus spp. are diploid (2n = 22), while S. megalanthus is a tetraploid (2n = 44). Irregular chromosome disjunction at anaphase I in pollen mother cells of S. megalanthus is probably the major cause of its reduced pollen viability a...

In this paper we present an extended formulation for the dominating set polytope via facility location. We show that with this formulation we may describe the dominating set polytope for some class of graphs as cacti graphs, though its description in the natural node variables dimension has been only partially obtained. Moreover, the inequalities describing this polytope have coefficients in {−...

The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We first review these functional equations, putting the emphasis on the structural relationships between the various classes of graphs. We then investigate the prob...

We endow the normalized cacti with structure of an ?-operad by showing that its existing composition laws are associative up to all higher homotopies. The homotopies encoded a new topological operad bracketed trees which we relate both enrichment dendroidal category ? and Boardman-Vogt W-construction on operads.

A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G) is the maximum number of colours used in a b-colouring of G. If m(G) is the largest integer k such that G has at least k vertices with degree at least k − 1, then we know that χb(G) ≤ m(G). Irving and Manlove [1] prove that, if T is...