In this note we construct nested partially balanced incomplete block designs based on \(NC_{m}\)-scheme. Secondly we construct NPBIB designs from a given PBIB design with \(\lambda_{1} = 1\) and \(\lambda_{2} = 0\) with same association scheme for both systems of PBIB designs. Finally, we give some results and examples where the two systems of PBIB designs in NPBIB designs have different association schemes.

This paper discusses the covering property and the Uniqueness Property of Minima (UPM) for linear forms in an arbitrary number of variables, with emphasis on the case of three variables (triple loop graph). It also studies the diameter of some families of undirected chordal ring graphs. We focus upon maximizing the number of vertices in the graph for given diameter and degree. We study the result in \([2]\), we find that the family of triple loop graphs of the form \(G(4k^2+2k+1; 1;2k+1; 2k^2)\) has a larger number of nodes for diameter \(k\) than the family \(G(3k^2+3k+1;1;3k+1;3k+2)\) given in \([2]\). Moreover, we show that both families have the Uniqueness Property of Minima.

In this paper, an algorithm based on. trades is presented to classify two classes of large sets of \(t\)-designs, namely \(LS[14](2, 5, 10)\) and \(LS[6](3, 5, 12)\).

In this work, we study which tubular surfaces verify that the embeddings of infinite, locally finite connected graphs without vertex accumulation points are embeddings without edge accumulation points. Furthermore, we characterize the graphs which admit embeddings with no edge accumulation points in the sphere with \(n\) ends in terms of forbidden subgraphs.

In this paper, self-centered, bi-eccentric splitting graphs are characterized. Further various bounds for domination number, global domination number and the neighborhood number of these graphs are obtained.

In this study we are going to give a new \((t,k)\)-geodetic set definition. This is a refinement of the geodetic set definition given in \([11]\). With this new definition we obtain more information about the graph. We also give a relationship between the \((t,k)\)-geodetic set and the integrity of a graph. By using a \((t,k)\)-geodetic set we give a new proof for the upper bound of integrity of trees and unicycle graphs.

For a long time we had thought that there does not exist an OGDD of type \(4^4\). In this article, an OGDD of type \(4^4\) will be constructed.

Consider a tree \(T = (V, E)\) with root \(r \in V\) and \(|V| = N\). Let \(p_v\) be the probability that a user wants to access node \(v\). A bookmark is an additional link from \(r\) to any other node of \(T\). We want to add \(k\) bookmarks to \(T\), so as to minimize the expected access cost from \(r\), measured by the average length of the shortest path. We present a characterization of an optimal assignment of \(k\) bookmarks in a perfect binary tree with uniform probability distribution of access and \(k \leq \sqrt{N + 1}\).

We show various combinatorial identities that are generated by tree counting arguments. In particular, we give formulas for \(n^p\) and \(\tau(K_{s,t})\) which establishes an equivalence.

The question of necessary and sufficient conditions for the existence of a simple \(3\)-uniform hypergraph with a given degree sequence is a long outstanding open question. We provide a result on degree sequences of \(3\)-hypergraphs which shows that any two \(3\)-hypergraphs with the same degree sequence can be transformed into each other using a sequence of trades, also known as null-\(3\)-hypergraphs. This result is similar to the Havel-Hakimi theorem for degree sequences of graphs.