**Volume 21** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On the inequalities for beta function**

*Original research paper. Pages 1—7*

Barkat Ali Bhayo and József Sándor

Full paper (PDF, 152 Kb) | Abstract

Here authors establish the sharp inequalities for classical beta function by studying the inequalities of trigonometric sine function.

**Some infinite series involving arithmetic functions**

*Original research paper. Pages 8—14*

Ramesh Kumar Muthumalai

Full paper (PDF, 172 Kb) | Abstract

Some identities for infinite series involving arithmetic functions are derived through Jacobi symbols (−1|*k*) and (2|*k*). Using these identities, some Dirichlet series are expressed in terms of Hurwitz zeta function.

**On some Pascal’s like triangles. Part 9**

*Original research paper. Pages 15—22*

Krassimir T. Atanassov

Full paper (PDF, 131 Kb) | Abstract

In a series of papers, Pascal’s like triangles with different forms have been described. Here, three-dimensional analogues of these triangles are given and some of their properties are studied.

**On some Pascal’s like triangles. Part 10**

*Original research paper. Pages 23—34*

Krassimir T. Atanassov

Full paper (PDF, 150 Kb) | Abstract

In a series of papers, Pascal’s like triangles with different forms have been described. Here, a new type of triangles is discussed. In the formula for their generation, operation summation is changed with operation subtraction. Some of their properties are studied.

**The Golden Ratio family and the Binet equation**

*Original research paper. Pages 35—42*

A. G. Shannon and J. V. Leyendekkers

Full paper (PDF, 101 Kb) | Abstract

The Golden Ratio can be considered as the first member of a family which can generate a set of generalized Fibonacci sequences. Here we consider some related problems in terms of the Binet form of these sequences, {*F*_{n}(*a*)}, where the sequence of ordinary Fibonacci numbers can be expressed as {*F*_{n}(5)} in this notation. A generalized Binet equation can predict all the elements of the Golden Ratio family of sequences. Identities analogous to those of the ordinary Fibonacci sequence are developed as extensions of work by Filipponi, Monzingo and Whitford in *The Fibonacci Quarterly*, by Horadam and Subba Rao in the *Bulletin of the Calcutta Mathematical Society*, within the framework of Sloane’s *Online Encyclopedia of Interger Sequences*.

**More new properties of modified Jacobsthal and Jacobsthal–Lucas numbers**

*Original research paper. Pages 43—54*

Julius Fergy T. Rabago

Full paper (PDF, 192 Kb) | Abstract

We present some new elementary properties of modified Jacobsthal (Atanassov, 2011)

and Jacobsthal–Lucas numbers (Shang, 2012).

**On right circulant matrices with general number sequence**

*Original research paper. Pages 55—58*

Aldous Cesar F. Bueno

Full paper (PDF, 140 Kb) | Abstract

In this paper, the elements of the general number sequence were used as entries for right circulant matrices. The eigenvalues, the Euclidean norm and the inverse of the resulting matrices were obtained.

**On directed pathos line cut vertex digraph of an arborescence**

*Original research paper. Pages 59—69*

Nagesh H. M. and R. Chandrasekhar

Full paper (PDF, 630 Kb) | Abstract

In this paper we define the digraph valued function (digraph operator), namely the line cut vertex digraph *n*(*D*) of a digraph *D* and the directed pathos line cut vertex digraph *DPn*(*T*) of an arborescence *T*. Planarity, outer planarity, maximal outer planarity, minimally non-outer planarity, and crossing number one properties of *DPn*(*T*) are discussed. Also, the problem of reconstructing an arborescence from its directed pathos line cut vertex digraph is presented.

**On Π**_{k}–connectivity of some product graphs

*Original research paper. Pages 70—79*

B. Chaluvaraju, Medha Itagi Huilgol, Manjunath N. and Syed Asif Ulla S.

Full paper (PDF, 278 Kb) | Abstract

Let *k* be a positive integer. A graph *G* = (*V*, *E*) is said to be Π*k*-connected if for any given subset *S* of *V*(*G*) with |*S*| = *k*, the subgraph induced by *S* is connected. In this paper, we consider Π_{k}–connected graphs under different graph valued functions. Π_{k}–connectivity of Cartesian product, normal product, join and corona of two graphs have been obtained in this paper.

**Embedding index in some classes of graphs**

*Original research paper. Pages 80—88*

M. Kamal Kumar and R. Murali

Full paper (PDF, 144 Kb) | Abstract

A Subset *S* of the vertex set of a graph *G* is called a dominating set of *G* if each vertex of *G* is either in *S* or adjacent to at least one vertex in *S*. A partition *D* = {*D*_{1}, *D*_{2}, …, *D*_{k}} of the vertex set of *G* is said to be a domatic partition or simply a *d*-partition of *G* if each class *D*_{i} of *D* is a dominating set in *G*. The maximum cardinality taken over all *d*-partitions of *G* is called the domatic number of *G* denoted by *d*(*G*). A graph *G* is said to be domatically critical or *d*-critical if for every edge *x* in *G*, *d*(*G* – *x*) < *d*(*G*), otherwise *G* is said to be domatically non *d*-critical. The embedding index of a non *d*-critical graph *G* is defined to be the smallest order of a *d*-critical graph *H* containing *G* as an induced subgraph denoted by *θ*(*G*) . In this paper, we find the *θ*(*G*) for the Barbell graph, the Lollipop graph and the Tadpole graph.

**Volume 21** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4