a graph that contains a hamiltonian cycle is called a hamiltonian graph. in this paper wecompute the first and the second geometric – arithmetic indices of hamiltonian graphs. thenwe apply our results to obtain some bounds for fullerene.

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

We give a bijection between permutations of 1, . . . , 2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate selecting from a set of 2n items: the permutation encodes the players’ preference ordering of the items, and the Dyck paths encode the order in which items are selected under optimal play. We enumerate permutatio...

The first case of recurrent neuritis was published by Eichhorst in 1890 (1). In 1958, Austin (2) described steroid responsiveness, recurrence and cerebrospinal fluid (CSF) protein elevation. Dyck and colleagues (3) described the clinicopathological findings of monophasic progression, steady deterioration, stepwise progression and recurrence, with histologic peripheral nervious system (PNS) infl...

We exploit Krattenthaler’s bijection between the set Sn(3-1-2) of permutations in Sn avoiding the classical pattern 3-1-2 and Dyck n-paths to study the distribution of every consecutive pattern of length 3 on the set Sn(3-1-2). We show that these consecutive patterns split into 3 equidistribution classes, by means of an involution on Dyck paths due to E.Deutsch. In addition, we state equidistri...

The paper formulates projective dependency grammars in terms of constraints on a string based encoding of dependency trees and develops an approach to obtain a regular approximation for these grammars. In the approach, dependency analyses are encoded with balanced bracketing that encodes dependency relations among the words of the analyzed sentence. The brackets, thus, indicate dependencies rat...

A possible upgrade of a curriculum in undergraduate course in combinatorics is presented by giving more bijective proofs in the standard (or primary) combinatorics and by adding some topics on more refined (or secondary) combinatorics, including Dyck and Motzkin paths, Catalan, Narayana and Motzkin numbers and secondary structures coming from biology. Some log–convexity properties and asymptoti...

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain st...

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic reflection groups. In particular, we explore Dyck paths of type B together with an area statistic and a major index. Then, we construct bijections between non-nesting and reverse non-crossing partitions for types A and B. These bijections simultaneously send the area statistic and the major index...