Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to \(\{\pm 1\}\), have been introduced as a way to construct multilevel zero-correlation zone sequences for use in approximately synchronized code division multiple access (AS-CDMA) systems. This paper provides a construction technique to produce \(2^m \times 2^m\) MHMs whose \(2^m\) alphabet entries form an arithmetic progression, up to sign. This construction improves upon existing constructions because it permits control over the spacing and overall span of the MHM entries. MHMs with such regular alphabets are a more direct generalization of traditional Hadamard matrices and are thus expected to be more useful in applications analogous to those of Hadamard matrices. This paper also introduces mixed-circulant MHMs which provide a certain advantage over known circulant MHMs of the same size.

MHMs over the Gaussian (complex) and Hamiltonian (quaternion) integers are introduced. Several constructions are provided, including a generalization of the arithmetic progression construction for MHMs over real integers. Other constructions utilize amicable pairs of MHMs and c-MHMs, which are introduced as natural generalizations of amicable orthogonal designs and c-Hadamard matrices, respectively. The constructions are evaluated against proposed criteria for interesting and useful MHMs over these generalized alphabets.

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\text{dim}(G)\) is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we show that the sunlet graphs, the rising sun graphs, and the co-rising sun graphs have constant metric dimension.

A sequence \(\{a_i : 1 \leq i \leq k\}\) of integers is a weak Sidon sequence if the sums \(a_i + a_j\) are all different for any \(i < j\). Let \(g(n)\) denote the maximum integer \(k\) for which there exists a weak Sidon sequence \(\{a_i : 1 \leq i \leq k\}\) such that \(1 \leq a_1 < \cdots < a_k \leq n\). Let the weak Sidon number \(G(k) = \text{min}\{n \mid g(n) = k\}\). In this note, \(g(n)\) and \(G(k)\) are studied, and \(g(n)\) is computed for \(n \leq 172\), based on which the weak Sidon number \(G(k)\) is determined for up to \(k = 17\).

In this paper, we show that there exist all admissible 4-GDDs of type \(g^6m^1\) for \(g \equiv 0 \pmod{6}\). For 4-GDDs of type \(g^u m^1\), where \(g\) is a multiple of 12, the most values of \(m\) are determined. Particularly, all spectra of 4-GDDs of type \(g^um^1\) are attained, where \(g\) is a multiple of 24 or 36. Furthermore, we show that all 4-GDDs of type \(g^um^1\) exist for \(g = 10, 20, 28, 84\) with some possible exceptions.

Let \( f(n) \) be the maximum number of edges in a graph on \( n \) vertices in which no two cycles have the same length. Erdős raised the problem of determining \( f(n) \). Erdős conjectured that there exists a positive constant \( c \) such that \( ex(n, C_{2k}) \geq cn^{1+\frac{1}{k}} \). Hajós conjectured that every simple even graph on \( n \) vertices can be decomposed into at most \(\frac{n}{2}\) cycles. We present the problems, conjectures related to these problems, and we summarize the known results. We do not think Hajós’ conjecture is true.

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard). More than one guard is allowed to move in response to an attack. The \( m \)-eternal domination number is the minimum number of guards needed to defend the graph. We characterize the trees achieving several upper and lower bounds on the \( m \)-eternal domination number.

Introduced in 1947, the Wiener index (sum of distances between all pairs of vertices) is one of the most studied chemical indices. Extensive results regarding the extremal structure of the Wiener index exist in the literature. More recently, the Gamma index (also called the Terminal Wiener index) was introduced as the sum of all distances between pairs of leaves. It is known that these two indices coincide in their extremal structures and that a nice functional relation exists for \(k\)-ary trees but not in general. In this note, we consider two natural extensions of these concepts, namely the sum of all distances between internal vertices (the Spinal index) and the sum of all distances between internal vertices and leaves (the Bartlett index). We first provide a characterization of the extremal trees of the Spinal index under various constraints. Then, its relation with the Wiener index and Gamma index is studied. The functional relation for \(k\)-ary trees also implies a similar result on the Bartlett index.

For an \( n \)-connected graph \( G \), the \( n \)-wide diameter \( d_n(G) \) is the minimum integer \( m \) such that for any two vertices \( x \) and \( y \) there are at least \( n \) internally disjoint paths of length at most \( m \) from \( x \) to \( y \). For a given integer \( l \), a subset \( S \) of \( V(G) \) is called a \( (l,n) \)-dominating set of \( G \) if for any vertex \( x \in V(G) – S \) there are at least \( n \) internally disjoint paths of length at most \( l \) from \( S \) to \( x \). The minimum cardinality among all \( (l,n) \)-dominating sets of \( G \) is called the \( (l,n) \)-domination number. In this paper, we obtain that the \( (l,\omega) \)-domination numbers of the circulant digraph \( G(d^n; \{1, d, \ldots, d^{n-1}\}) \) is equal to 2 for \( 1 \leq \omega \leq n \) and \( d_\omega(G) – (g(d,n) + \delta) \leq l \leq d_\omega(G) – 1 \), where \( g(d,n) = \text{min} \{e\lceil \frac{n}{2} \rceil – e – 2, (\lfloor \frac{n}{2} \rfloor + 1)(e – 1) – 2\} \), \( \delta = 0 \) for \( 1 \leq \omega \leq n – 1 \) and \( \delta = 1 \) for \( \omega = n \).