It is well-known that typical properties of random graphs can be derived from the q → 1 limit of the Potts model of statistical mechanics. We show that, additionally, the Legendre transform of the Potts free energy with respect to ln q allows to study large deviation properties of the random graph ensemble characterizing graphs with an atypical number of components. We also demonstrate how the ...

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yield...

An equitable vertex coloring for the splitting of block circulant graphs is investigated. The comprises matrices, where each itself a matrix. These blocks in row are cyclically shifted one place to right from those previous row. We approached such matrix representation and derived their independent sets using neighbourhoods vertex. This classification makes process be simpler most cases. In thi...

In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.

Secondary structures of polynucleotides can be view as a certain class of planar vertex-labeled graphs. We construct recursion formulae enumerating various sub-classes of these graphs as well as certain structural elements (sub-graphs). First order asymptotics are derived and their dependence on the logic of base pairing is computed and discussed.

The generalized hierarchical product of graphs was introduced by L. Barriére et al in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb latt...

In this paper we study proximity structures like Delaunay triangulations based on geometric graphs, i.e. graphs which are subgraphs of the complete geometric graph. Given an arbitrary geometric graph G, we define several Voronoi diagrams, Delaunay triangulations, relative neighborhood graphs, Gabriel graphs and then study their complexities when G is a general geometric graph or G is some speci...

We count labelled chordal graphs with no induced path of length 3, both exactly and asymptotically. These graphs correspond to rooted trees in which no vertex has exactly one child, and each vertex has been expanded to a clique. Some properties of random graphs of this type are also derived. The corresponding unlabelled graphs are in 1-1 correspondence with unlabelled rooted trees on the same n...

An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to graphs with edge-colorings by two colors, and give an excluded-subgraph characterization for graphs with such orderings. As an application, we announce some forthcoming resul...

in this paper we define the removable cycle that, if $im$ is a class of graphs, $gin im$, the cycle $c$ in $g$ is called removable if $g-e(c)in im$. the removable cycles in eulerian graphs have been studied. we characterize eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for eulerian graph to have removable cycles h...