The chromatic polynomial of a graph \( G \), \( P(G; \lambda) \), is the polynomial in \( \lambda \) which counts the number of distinct proper vertex \( \lambda \)-colorings of \( G \), given \( \lambda \) colors. We compute \( P(C_4 \times P_n; \lambda) \) and \( P(C_5 \times P_n; \lambda) \) in matrix form and will find the generating function for each of these sequences.

The \( n \)-cube is the graph whose vertices are all binary words of length \( n > 1 \) and whose edges join vertices that differ in exactly one entry; i.e., are at Hamming distance \( 1 \) from each other. If a word has a non-empty prefix, not the entire word, which is also a suffix, then it is said to be bordered. A word that is not bordered is unbordered. Unbordered words have been studied extensively and have applications in synchronizable coding and pattern matching. The neighborhood of an unbordered word \( w \) is the word itself together with the set of words at Hamming distance \( 1 \) from \( w \). Over the binary alphabet, the neighborhood of an unbordered word \( w \) always contains two bordered words obtained by complementing the first and last entries of \( w \). We determine those unbordered words \( w \) whose neighborhoods otherwise contain only unbordered words.

Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A labeling \( f : V \to \{0,1\} \) induces a partial edge labeling \( f^* : E \to \{0,1\} \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{|f^{*-1}(0) – f^{*-1}(1)| : |f^{-1}(0) – f^{-1}(1)| \leq 1\} \). In this paper, we study the balance index sets of graphs which are \( L \)-products with cycles and complete graphs.

For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) between \( u \) and \( v \) is the length of a longest \( u – v \) path in \( G \). The detour diameter \( \text{diam}_D(G) \) of \( G \) is the greatest detour distance between two vertices of \( G \). Two vertices \( u \) and \( v \) are detour antipodal in \( G \) if \( D(u,v) = \text{diam}_D(G) \). The detour antipodal graph \( \text{DA}(G) \) of a connected graph \( G \) has the same vertex set as \( G \) and two vertices \( u \) and \( v \) are adjacent in \( \text{DA}(G) \) if \( u \) and \( v \) are detour antipodal vertices of \( G \). For a connected graph \( G \) and a nonnegative integer \( r \), define \( \text{DA}^r(G) \) as \( G \) if \( r = 0 \) and as the detour antipodal graph of \( \text{DA}^{r-1}(G) \) if \( r > 0 \) and \( \text{DA}^{r-1}(G) \) is connected. Then \( \{\text{DA}^r(G)\} \) is the detour antipodal sequence of \( G \). A graph \( H \) is the limit of \( \{\text{DA}^r(G)\} \) if there exists a positive integer \( N \) such that \( \text{DA}^r(G) \cong H \) for all \( r \geq N \). It is shown that \( \{\text{DA}^r(G)\} \) converges if \( G \) is Hamiltonian. All graphs that are the limit of the detour antipodal sequence of some Hamiltonian graph are determined.

For a vertex \( x \) in a graph \( G \), we define \( \Psi_1(x) \) to be the number of edges in the closed neighborhood of \( x \). Vertex \( x^* \) is a neighborhood champion if \( \Psi_1(x^*) > \Psi_1(x) \) for all \( x \neq x^* \). We also refer to such an \( x^* \) as a unique champion. For \( d \geq 4 \), let \( n_0(1,d) \) be the smallest number such that for every \( n \geq n_0(1,d) \) there exists an \( n \)-vertex \( d \)-regular graph with a unique champion. Our main result is that \( n_0(1,d) \) satisfies \( d+3 \leq n_0(1,d) < 3d+1 \). We also observe that there can be no unique champion vertex when \( d = 3 \).

In this paper, we consider the non-existence of some bi-level orthogonal arrays (O-arrays) of strength six, with \( m \) constraints (\( 6 \leq m \leq 32 \)), and with index set \( \mu \) (\( 1 \leq \mu \leq 512 \)). The results presented here tend to improve upon the results available in the literature.

We present constructions and results about GDDs with two groups and block size five in which each block has configuration \((s, t)\), that is, in which each block has exactly \(s\) points from one of the two groups and \(t\) points from the other. After some results for a general \(k\), \(s\), and \(t\), we consider the \((2,3)\) case for block size \(5\). We give new necessary conditions for this family of GDDs and give minimal or near-minimal index examples for all group sizes \(n \geq 4\) except for \(n = 24s + 17\).

We compute the limiting average connectivity \(\overline{\kappa}\) of the family of \(3\)-regular expander graphs whose members are formed from the finite fields \(\mathbb{Z}_p\), by connecting every \(x \in \mathbb{Z}_p\) with \(x\pm1\) and \(x^{-1}\), all computations performed modulo \(p\). Namely, we show

\[\lim_{p\to\infty} \overline{\kappa}(\mathbb{Z}_p) = 3\]

for primes \(p\). We compare this behavior with an upper bound on the expected value of \(\overline{\kappa}(\mathbb{Z}_n)\) for a more general class \(\{\mathbb{Z}_n\}_{n\in\mathbb{N}}\) of related graphs.