The number \( g_3^{(4)}(v) \) represents the minimum cardinality of a pairwise balanced design on \( v \) elements in which the largest block size is four and every pair occurs exactly three times. We give a survey of the results for this quantity.

Let \( G_1 \) and \( G_2 \) be any two 2-regular graphs, each with \( n \) vertices. Let \( G \) be any cubic graph obtained from \( G_1 \) and \( G_2 \) by adding \( n \) edges, each of which joins a vertex in \( G_1 \) to a vertex in \( G_2 \). We show that \( G \) has a myriad of vertex-magic total labelings, with at least three different magic constants. This class of cubic graphs includes all generalized Petersen graphs.

Let [n, k, d]_q codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). In this paper, the existence of the following codes is proven: [42, 6, 30]₈, [49, 6, 36]₈, [78, 6, 60]₈, [84, 6, 65]₈, [91, 6, 71]₈, [96, 6, 75]₈, [102, 6, 80]₈, [108, 6, 85]₈, [114, 6, 90]₈,and  [48, 6, 35]₉, [54, 6, 40]₉, [60, 6, 45]₉, [96, 6, 75]₉, [102, 6, 81]₉, [108, 6, 85]₉, [114, 6, 90]₉, [126, 6, 100]₉, [132, 6, 105]₉. The nonexistence of five codes over GF(9) is also proven. All of these results improve the respective upper and lower bounds in Brouwer’s table [2].

In this paper, we obtain some necessary existence conditions for bi-level balanced arrays of strength six by using some classical inequalities and by expressing the moments of the weights of the columns of such arrays in terms of its parameters. We present some illustrative examples to compare these results with the earlier known results.

This paper describes a comprehensive approach to the analysis and synthesis of tree-structured communication networks. First, a class of models for tree-structured communication networks is proposed. Then, performance parameters such as communication delays and network reliability are defined, and efficient algorithms for calculating these parameters are provided. Subsequently, an application of a powerful tree-generating algorithm to the synthesis of optimal communication networks is described. The universal approach of this algorithm allows for its use in conjunction with the proposed model and the algorithms for calculating values of performance parameters. The paper shows sample optimal tree-structured networks resulting from applying the synthesis algorithm for various optimization parameters.

A vertex \( v \) of a connected graph \( G \) is an eccentric vertex of a vertex \( u \) if \( v \) is a vertex at greatest distance from \( u \); while \( v \) is an eccentric vertex of \( G \) if \( v \) is an eccentric vertex of some vertex of \( G \). The subgraph of \( G \) induced by its eccentric vertices is the eccentric subgraph of \( G \).

A vertex \( v \) of \( G \) is a boundary vertex of a vertex \( u \) if \( d(u,w) \leq d(u,v) \) for each neighbor \( w \) of \( v \). A vertex \( v \) is a boundary vertex of \( G \) if \( v \) is a boundary vertex of some vertex of \( G \). The subgraph of \( G \) induced by its boundary vertices is the boundary of \( G \). A vertex \( v \) is an interior vertex of \( G \) if for every vertex \( u \) distinct from \( v \), there exists a vertex \( w \) distinct from \( v \) such that \( d(u,w) = d(u,v) + d(v,w) \). The interior of \( G \) is the subgraph of \( G \) induced by its interior vertices. A vertex \( v \) is a boundary vertex of a connected graph if and only if \( v \) is not an interior vertex. For every graph \( G \), there exists a connected graph \( H \) such that \( G \) is both the center and interior of \( H \).

Relationships between the boundary and the periphery, center, and eccentric subgraph of a graph are studied. The boundary degree of a vertex \( v \) in a connected graph \( G \) is the number of vertices \( u \) in \( G \) having \( v \) as a boundary vertex. We study, for each pair \( r,n \) of integers with \( r \geq 0 \) and \( n \geq 3 \), the existence of a connected graph \( G \) of order \( n \) such that every vertex of \( G \) has boundary degree \( r \). We also study the boundary vertices of a connected graph from different points of view.

Let \( G \) be a simple graph having a maximum matching \( M \). The deficiency \( \text{def}(G) \) of \( G \) is the number of vertices unsaturated by \( M \). A bridge in a connected graph \( G \) is an edge \( e \) of \( G \) such that \( G-e \) is disconnected. A graph is said to be almost cubic (or almost 3-regular) if one of its vertices has degree \( 3 + e \), \( e \geq 0 \), and the others have degree 3. In this paper, we find the minimum number of bridges of connected almost cubic graphs with a given deficiency.

The cardinality of the minimal pairwise balanced designs on \( v \) elements with largest block size \( k \) is denoted by \( g^{(k)}(v) \). It is known that \(30 \leq g^{(4)}(18) \leq 33.\)In this note, we show that \(31 \leq g^{(4)}(18).\)

In this paper, we introduce two new classes of critical sets, \( t \)-uniform and \( T \)-uniform (where \( t \) is a positive integer and \( T \) is a partial Latin square). We identify, up to isomorphism, all \( t \)-uniform critical sets of order \( n \), where \( 2 \leq n \leq 6 \). We show that the completable product of two \( T \)-uniform critical sets is a \( T \)-uniform critical set for certain partial Latin squares \( T \), and then apply this theorem to small examples to generate infinite families of \( T \)-uniform critical sets.