In a recent paper, E. Gelman provided an exact formula for the number of subgroups of a given index for the Baumslag-Solitar groups \( \text{BS}(p, q) \) when \( p \) and \( q \) are coprime. We use Gelman’s proof as the basis for an algorithm that computes a maximal set of inequivalent permutation representations of \( \text{BS}(p, q) \) with degree \( n \). The computational complexity of this algorithm is linear in both space and time with respect to the index. We compare the performance of this algorithm with the Todd-Coxeter procedure, which generally lacks a polynomial bound on the number of cosets used during the enumeration process.

An \( H_2 \) graph is a multigraph on three vertices with a double edge between one pair of distinct vertices and a single edge between the other two pairs. The problem of decomposing a complete multigraph \( 3K_{8t} \) into \( H_2 \) graphs has been completely solved. In this paper, we describe some new procedures for such decompositions and pose the following question: Can these procedures be adapted or extended to provide a unified proof of the existence of \( H_2(8t, \lambda) \)’s?

For a nontrivial connected graph \( G \) of order \( n \) and a cyclic ordering \( s: v_1, v_2, \ldots, v_n, v_{n+1} = v_1 \) of \( V(G) \), let \( d(s) = \sum_{i=1}^n d(v_i, v_{i+1}) \), where \( d(v_i, v_{i+1}) \) is the distance between \( v_i \) and \( v_{i+1} \) for \( 1 \leq i \leq n \). The Hamiltonian number \( h(G) \) and the upper Hamiltonian number \( h^+(G) \) of \( G \) are defined as:

1. \( h(G) = \min \{ d(s) \} \), where the minimum is taken over all cyclic orderings \( s \) of \( V(G) \).
2. \( h^+(G) = \max \{ d(s) \} \), where the maximum is taken over all cyclic orderings \( s \) of \( V(G) \).

All connected graphs \( G \) with \( h^+(G) = h(G) \) and \( h^+(G) = h(G) + 1 \) have been characterized in [6, 13]. In this note, we first present a new and significantly improved proof of the characterization of all graphs whose Hamiltonian and upper Hamiltonian numbers differ by 1. We then determine all pairs of integers that can be realized as the order and upper Hamiltonian number of some tree.

For natural numbers \( n \) and \( k \), where \( n > 2k \), a generalized Petersen graph \( P(n,k) \) is obtained by letting its vertex set be \( \{u_1, u_2, \ldots, u_n\} \cup \{v_1, v_2, \ldots, v_n\} \) and its edge set be the union of \( \{u_i u_{i+1}, u_i v_i, v_i v_{i+k}\} \) over \( 1 \leq i \leq n \), where subscripts are reduced modulo \( n \). In this paper, an integer programming formulation for Roman domination is established, which is used to give upper bounds for the Roman domination numbers of the generalized Petersen graphs \( P(n,3) \) and \( P(n,4) \). Together with the dynamic algorithm, we determine the Roman domination number of the generalized Petersen graph \( P(n,3) \) for \( n \geq 5 \).

In this paper, we introduce the zero-divisor graph \(\Gamma(L)\) of a meet-semilattice \(L\) with 0. It is shown that \(\Gamma(L)\) is connected with \(\text{diam}(\Gamma(L)) \leq 3\) and if \(\Gamma(L)\) contains a cycle, then the core \(K\) of \(\Gamma(L)\) is a union of 3-cycles and 4-cycles.

A unit distance graph is a finite simple graph which may be drawn on the plane so that its vertices are represented by distinct points and the edges are represented by closed line segments of unit length. In this paper, we show that the only primitive strongly regular unit distance graphs are \((i)\) the pentagon, \((ii)\) \(K_3 \times K_3\), \((iii)\) the Petersen graph, and \((iv)\) possibly the Hoffman-Singleton graph.

Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular symplectic spaces over finite fields.

In \(1969\), Dewdney introduced the set \(\Gamma\) of \({primal \;graphs}\), characterized by the following two properties: every finite, simple graph \(G\) is the union of non-isomorphic, edge-disjoint subgraphs of \(G\) so that each of the subgraphs is in \(\Gamma\); and, if \(G\) is in \(\Gamma\), then the only such union consists of \(G\) itself. In the period around 1990, several works concerning the determination of the graphs in \(\Gamma\) were published and one Ph.D. thesis written. However, the classification of the members of \(\Gamma\) remains elusive. The main point of this work is to simplify and unify some of the principal results of Preen’s Ph.D. thesis that generalize earlier results about primal graphs with maximum degree \(2\).

In order to characterize convex polyhedra with regular polygonal faces by a minimal number of parameters, we first introduce some new parameters, then we analyze a table of their values to see how well different sets of parameters tell these solids apart, and finally we present their characterization by four parameters.

In this paper, we show a short proof of the \( q \)-binomial theorem by Schützenberger’s identity with \( q \)-commuting variables.