This paper discusses the permutations that are generated by rotating \(k \times k\) blocks of squares in a union of overlapping \(k \times (k + 1)\) rectangles. It is found that the single-rotation parity constraints effectively determine the group of accessible permutations. If there are \(m\) squares, and the space is partitioned as a checkerboard with \(m\) squares shaded and \(n – m\) squares unshaded, then the four possible cases are \(A_n\), \(S_n\), \(A_m \times A_{n-m}\), and the subgroup of all even permutations in \(S_m \times S_{n-m}\), with exceptions when \(k = 2\) and \(k = 3\).

The packing and covering numbers for the 4-stars were determined by Roditty in 1986. In this paper, we improve and extend these results by finding a corresponding maximum packing and minimum covering of the complete graph with 4-stars for every possible leave graph and excess graph.

We examine the Borda voting method, which has numerous interesting mathematical properties. We determine when a candidate can win a Borda election with all \(i\)th place votes and present a method of constructing ballots that yield such a victory. Then we present a connection between Borda elections and semi-magic squares. We show how a Borda election result gives rise to a semi-magic square, and we show that given any semi-magic square there exists at least one Borda election result corresponding to it.

In 2000, Rees and Shalaby constructed simple indecomposable two-fold cyclic triple systems for all \(v \equiv 0, 1, 3, 4, 7, \text{ and } 9 \pmod{12}\) where \(v = 4\) or \(v \geq 12\), using Skolem-type sequences.
We construct, using Skolem-type sequences, three-fold triple systems having the properties of being cyclic, simple, and indecomposable for all admissible orders \(v\), with some possible exceptions for \(v = 9\) and \(v = 24c + 57\), where \(c \geq 2\) is a constant.

Counting the number of maximal independent sets is \#P-complete even for chordal graphs. We prove that the number of maximal independent sets in a subclass \(\mathcal{G}_n^R\) (Right power set graphs) of chordal graphs can be computed in polynomial time using Golomb’s nonlinear recurrence relation. We provide a recursive construction of \(\mathcal{G}_n^R\) and prove that there are \(2^\frac{|V(\mathcal{G}_n^R)|+1}{4}\) maximum independent sets in \(\mathcal{G}_n^R\). We also provide a polynomial-time algorithm to solve the maximum independent set problem (MISP) in a superclass \(\mathcal{F}_n\) of the complement of \(\mathcal{G}_n^R\).

The eccentric connectivity index of the molecular graph \(G\) was proposed by Sharma, Goswami, and Madan in 1997 \cite{17}. This index is defined as \(\xi^c(G) = \sum_{v\in V(G)} \deg(v) \, ec(v)\), where \(\deg(v)\) is the degree of vertex \(v\) in \(G\) and eccentricity \(ec(v)\) is the largest distance between \(v\) and any other vertex of \(G\). Thus, in this paper, we established the general formulas for the eccentric connectivity index of joining a special graph to its paths and of joining two different graphs by a path. Proofs were also provided.

We present a design for a seven-game tournament of the 7-player board game \emph{Diplomacy}, in which each player plays each country one time and each pair of players shares a border either 4 or 5 times. It is impossible for each pair of players to share a border the same number of times in such a tournament, and so the tournament presented is the most “balanced” possible in this sense. A similarly balanced tournament can be constructed for a generalized version of the game involving an arbitrary number of countries. We also present an infinite family of graphs that cannot be balanced.

A graph \(G = (V, E)\) with \(p\) vertices and \(q\) edges is called a Harmonic mean graph if it is possible to label the vertices \(v \in V\) with distinct labels \(f(v)\) from \(1, 2, \dots, q+1\) in such a way that when each edge \(e = uv\) is labeled with \(f(e = uv) = \left\lceil\frac{2f(u)f(v)}{f(u) + f(v)}\right\rceil\) or \(\left\lfloor \frac{2f(u)f(v)}{f(u) + f(v)} \right\rfloor\), then the edge labels are distinct. In this case, \(f\) is called a Harmonic mean labeling of \(G\). In this paper, we investigate some new families of Harmonic mean graphs.

The clique sum \(\Sigma = G[G_1,G_2,\ldots,G_n]\) is the lexicographic sum over \(G\) where each fiber \(G_i\) is a clique. We show the reconstruction number of \(\Sigma\) is three unless \(\Sigma\) is vertex transitive and \(G\) has order at least two. In the latter case, it follows that \(\Sigma = G[K_m]\) is a lexicographic product, and the reconstruction number is \(m+2\). This complements the bounds of Brewster, Hahn, Lamont, and Lipka. It also extends the work of Myrvold and Molina.

We provide a new proof of a result of Hanson and Toft classifying the maximum-size \(K_r\)-free graphs on \(n\) vertices with chromatic number at least \(r\).