Let \(p\) be a prime number and let \(\mathbb{F}_p\) be a finite field. In the first section, we give some preliminaries from elliptic curves over finite fields. In the second section, we consider the rational points on the elliptic curves \(E_{p,\lambda} : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{F}_p\) for primes \(p \equiv 3 \pmod{4}\), where \(\lambda \neq 0, 1\). We prove that the order of \(E_{p,\lambda}\) over \(\mathbb{F}_p\) is \(p+1\) if \(\lambda = 2,\frac{p+1}{2}\) or \(p-1\). Later, we generalize this result to \(\mathbb{F}_{p^n}\) for any integer \(n \geq 2\). Also, we obtain some results concerning the sum of \(x\)- and \(y\)-coordinates of all rational points \((x,y)\) on \(E_{p,\lambda}\) over \(\mathbb{F}_p\). In the third section, we consider the rank of \(E_\lambda : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{Q}\).

For over a decade, there has been considerable research on codes over \(\mathbb{Z}_4\) and other rings. In spite of this, no tables or databases exist for codes over \(\mathbb{Z}_4\), as is the case with codes over finite fields. The purpose of this work is to contribute to the creation of such a database. We consider cyclic, negacyclic and quasi-twisted \((QT)\) codes over \(\mathbb{Z}_4\). Some of these codes have binary images with better parameters than the best-known binary linear codes. We call such codes “good codes”. Among these are two codes which improve the bounds on the best-known binary non-linear codes. Tables of best cyclic and \(QT\) codes over \(\mathbb{Z}_4\) are presented.

Acharya and Hegde have introduced the notion of strongly \(k\)-indexable graphs: A \((p,q)\)-graph \(G\) is said to be strongly \(k\)-indexable if its vertices can be assigned distinct integers \(0,1,2,\ldots,p-1\) so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices can be arranged as an arithmetic progression \(k,k+1,k+2,\ldots,k+(q-1)\). Such an assignment is called a strongly \(k\)-indexable labeling of \(G\). Figueroa-Centeno et al. have introduced the concept of super edge-magic deficiency of graphs: Super edge-magic deficiency of a graph \(G\) is the minimum number of isolated vertices added to \(G\) so that the resulting graph is super edge-magic. They conjectured that the super edge-magic deficiency of the complete bipartite graph \(K_{m,n}\) is \((m-1)(n-1)\) and proved it for the case \(m=2\). In this paper, we prove that the conjecture is true for \(m=3,4,5\), using the concept of strongly \(k\)-indexable labelings \(^1\).

Let \(M = \{v_1, v_2, \ldots, v_t\}\) be an ordered set of vertices in a graph \(G\). Then \((d(u, v_1), d(u, v_2), \ldots, d(u, v_\ell))\) is called the \(M\)-location of a vertex \(u\) of \(G\). The set \(M\) is called a locating set if the vertices of \(G\) have distinct \(M\)-locations. A minimum locating set is a set \(M\) with minimum cardinality. The cardinality of a minimum locating set of \(G\) is called the Location Number \(L(G)\). This concept has wide applications in motion planning and in the field of robotics. In this paper, we consider networks with a binary tree as an underlying structure and determine the minimum locating set of such architectures. We show that the location number of an \(n\)-level \(X\)-tree lies between \(2^{n-3}\) and \(2^{n – 3} + 2\). We further prove that the location number of an \(N \times N\) mesh of trees is greater than or equal to \(N/2\) and less than or equal to \(N\).

In this paper, we give generalizations of Padovan numbers and Perrin numbers. We apply these generalizations for counting of special subsets of the set of \(n\) integers. Next, we give their graph representations with respect to the number of maximal \(k\)-independent sets in graphs.

In this paper, we show that the crossing number of the complete multipartite graph \(K_{1,1,3,n}\) is

\[\operatorname{cr}(K_{1,1,3,n}) = 4\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor + \lfloor\frac{3n}{2}\rfloor\]

Our proof depends on Kleitman’s results for the complete bipartite graphs [D. J. Kleitman, The crossing number of \(K_{5,n}\), J. Combin.Theory, \(9 (1970), 315-323\)]..

A near-perfect matching is a matching saturating all but one vertex in a graph. In this note, it is proved that if a graph has a near-perfect matching then it has at least two, moreover, a concise structure construction for all graphs with exactly two near-perfect matchings is given. We also prove that every connected claw-free graph \(G\) of odd order \(n\) (\(n \geq 3\)) has at least \(\frac{n+1}{2}\) near-perfect matchings which miss different vertices of \(G\).

le of an edge-coloured graph \(G^*\) such that there is no finite integer \(n\) for which it is possible to decompose \(rK_n^*\) into edge-disjoint colour-identical copies of \(G^*\). We investigate the problem of determining precisely when an edge-coloured graph \(G^*\) with \(r\) colours admits a \(G^*\)-decomposition of \(rK_n^*\), for some finite \(n\). We also investigate conditions under which any partial edge-coloured \(G^*\)-decomposition of \(rK_n^*\) has a finite embedding.

Four new combinatorial identities involving certain generalized \(F\)-partition functions and \(n\)-colour partition functions are proved bijectively. This leads to new combinatorial interpretations of four mock theta functions of S.Ramanujan.

In this paper, we introduce some contractive conditions of Meir-Keeler type for a pair of mappings, called MK-pair and L-pair, in the framework of cone metric spaces. We prove theorems which assure the existence and uniqueness of common fixed points for MK-pairs and L-pairs. As an application, we obtain a result on the common fixed point of a p-MK-pair, a mapping, and a multifunction in complete cone metric spaces. These results extend and generalize well-known comparable results in the literature.