I. Anderson and L. Ellison [7] demonstrated the existence of \( Z \)-cyclic Directed Triplewhist Tournament Designs and \( Z \)-cyclic Ordered Triplewhist Tournament Designs for all primes \( p \equiv 9 \pmod{16} \). It is shown here that their methodology can be generalized completely to deal with primes of the form \( p \equiv (2^k + 1) \pmod{2^{k+1}} \), \( k \geq 4 \).

An \( \mathbb{L}(n,d) \) is a linear space with constant point degree \( n+1 \), lines of size \( n \) and \( n-d \), and with \( v = n^2 – d \) points. Denote by \( b = n^2 + n + z \) the number of lines of an \( \mathbb{L}(n,d) \), then \( z \geq 0 \) and examples are known only if \( z = 0, 1 \) [7]. The linear spaces \( \mathbb{L}(n, d) \) were introduced in [7] in relation with some classification problems of finite linear spaces. In this note, starting from the symmetric configuration \( 45_7 \) of Baker [1], we give an example of \( \mathbb{L}(n,d) \), with \( n=7, d=4 \) and \( z=4 \).

We propose a multilevel cooperative search algorithm to compute upper bounds for \( C_\lambda(v,k,t) \), the minimum number of blocks in a \( t-(v,k,\lambda) \) covering design. Multilevel cooperative search is a search heuristic combining cooperative search and multilevel search. We first introduce a coarsening strategy for the covering design problem which defines reduced forms of an original \( t-(v,k,\lambda) \) problem for each level of the multilevel search. A new tabu search algorithm is introduced to optimize the problem at each level. Cooperation operators between tabu search procedures at different levels include new re-coarsening and interpolation operators. We report the results of tests that have been conducted on \( 158 \) covering design problems. Improved upper bounds have been found for \( 34 \) problems, many of which exhibit a tight gap. The proposed heuristic appears to be a very promising approach to tackle other similar optimization problems in the field of combinatorial design.

A Steiner tree for a set \( S \) of vertices in a connected graph \( G \) is a connected subgraph of \( G \) of smallest size that contains \( S \). The Steiner interval \( I(S) \) of \( S \) is the union of all vertices of \( G \) that belong to some Steiner tree for \( S \). A graph is strongly chordal if it is chordal and has the property that every even cycle of length at least six has an odd chord. We develop an efficient algorithm for finding Steiner intervals of sets of vertices in strongly chordal graphs.

A legal placement of Queens is any placement of Queens on an order \(N\) chessboard in which any two attacking Queens can be separated by a Pawn. The Queens’ independence separation number is the minimum number of Pawns which can be placed on an \(N \times N\) board to result in a separated board on which a maximum of \(m\) independent Queens can be placed. We prove that \(N + k\) Queens can be separated by \(k\) Pawns for large enough \(N\) and provide some results on the number of fundamental solutions to this problem. We also introduce separation relative to other domination-related parameters for Queens, Rooks, and Bishops.

In this paper, we describe two algorithms to identify the repeating subwords in a given partial word \( w_o = w_0[1,…,n] \). The first algorithm uses the suffix tree and the second algorithm uses the valency tree. Both algorithms take linear time to identify the repeating subwords of a partial word.

We present a class of Coded Petri net languages and study some algebraic properties. The purpose of introduction of this language is to bring out its usefulness in learning theory. We introduce an algorithm for learning a finite coded Petri net language and its running time is bounded by a polynomial function of given inputs.

In this present investigation, the authors obtain Fekete-Szegő’s inequality for certain normalized analytic functions \( f(z) \) defined on the open unit disk. As a special case of this result, Fekete-Szegő’s inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to give a generalization of the Fekete-Szegő inequalities obtained by Srivastava and Mishra and Ma and Minda.

This paper is mainly devoted to generate (special) (super) edge-magic labelings of graphs using matrices. Matrices are used in order to find lower bounds for the number of non-isomorphic (special) (super) edge-magic labelings of certain types of graphs. Also, new applications of graph labelings are discussed.

A well-designed interconnection network makes efficient use of scarce communication resources and is used in systems ranging from large supercomputers to small embedded systems on a chip. This paper deals with certain measures of vulnerability in interconnection networks. Let \( G \) be a non-complete connected graph and for \( S \subseteq V(G) \), let \( \omega(G – S) \) and \( m(G – S) \) denote the number of components and the order of the largest component in \( G – S \), respectively. The vertex-integrity of \( G \) is defined as

\[I(G) = \text{min}\{|S| + m(G – S) : S \subseteq V(G)\}.\]

A set \( S \) is called an \( I \)-set of \( G \) if \( I(G) = |S| + m(G – S) \). The rupture degree of \( G \) is defined by

\[r(G) = \text{max}\{\omega(G – S) – |S| – m(G – S) : S \subseteq V(G), \omega(G – S) \geq 2\}.\]

A set \( S \) is called an \( R \)-set of \( G \) if \( r(G) = \omega(G – S) – |S| – m(G – S) \). In this paper, we compute the rupture degree of complete binary trees and a class of meshes. We also study the relationship between an \( I \)-set and an \( R \)-set and find an upper bound for the rupture degree of Hamiltonian graphs.