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.

A non-empty \( r \)-element subset \( A \) of an \( n \)-element set \( X_n \), and a partition \( \pi \) of \( X_n \), are said to be orthogonal if every class of \( \pi \) meets \( A \) in exactly one element. A partition type is determined by the number of classes of each distinct size of the partition. The Johnson graph \( J(n,r) \) is the graph whose vertices are the \( r \)-element subsets of \( X_n \), with two sets being adjacent if they intersect in \( r-1 \) elements. A partition of a given type \( \tau \) is said to be a \( \tau \)-label for an edge \( AB \) in \( J(n,r) \) if the sets \( A \) and \( B \) are orthogonal to the partition. A cycle \( \mathcal{H} \) in the graph \( J(n,r) \) is said to be \( \tau \)-labeled if for every edge of \( \mathcal{H} \), there exists a \( \tau \)-label, and the \( \tau \)-labels associated with distinct edges are distinct. Labeled Hamiltonian cycles are used to produce minimal generating sets for transformation semigroups. We identify a large class of partition types \( \tau \) with a non-zero gap for which every Hamiltonian cycle in the graph \( J(n,r) \) can be \( \tau \)-labeled, showing, for example, that this class includes all the partition types with at least one class of size larger than 3 or at least three classes of size 3.