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.

The main result: If the vertices of a connected graph are labelled by positive real numbers such that the number assigned to any vertex is half of the sum of the numbers assigned to the vertices of its neighbourhood, then each label is an integral multiple of the minimum of all labels. Using this, a result proved earlier in [7] is derived: If \(V\) is a linearly dependent subset of a root system in which all roots have the same norm, then one of the roots in \(V\) is an integral combination of the other roots in \(V\).

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is said to be a dominating set of \(G\) if each \(v \in V – D\) is adjacent to at least one vertex of \(D\). The minimum cardinality of a dominating set of \(G\) is called the domination number of \(G\) and is denoted by \(\gamma(G)\). A dominating set \(D\) with cardinality \(\gamma(G)\) is called a \(\gamma\)-set of \(G\). Given a graph \(G\), a new graph, denoted by \(\gamma.G\) and called the \(\gamma\)-graph of \(G\), is defined as follows: \(V(\gamma.G)\) is the set of all \(\gamma\)-sets of \(G\) and two sets \(D\) and \(S\) of \(V(\gamma.G)\) are adjacent in \(\gamma.G\) if and only if \(|D \cap S| = \gamma(G) – 1\). A graph \(G\) is said to be \(\gamma\)-connected if \(\gamma.G\) is connected. A graph \(G\) is said to be a \(\gamma\)-graph if there exists a graph \(H\) such that \(\gamma-H\) is isomorphic to \(G\). In this paper, we show that trees and unicyclic graphs are \(\gamma\)-graphs. Also, we obtain a family of graphs which are not \(\gamma\)-graphs.

A Cayley graph is a graph constructed out of a group \(\Gamma\) and its generating set \(A\). In this paper, we determine the independent domination number, perfect domination number, and independent dominating sets of \(Cay(\mathbb{Z}_n, A)\), for a specified generating set \(A\) of \(\mathbb{Z}_n\).