It is well known that apart from the Petersen graph, there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and \(\delta\) vertices less than the Moore bound, where \(\delta\) is odd. Additionally, it is known that there exist only three graphs of maximum degree 3 and 2 vertices less than the Moore bound. In this paper, we consider graphs of maximum degree 3, diameter \( D \geq 2 \), and 4 vertices less than the Moore bound, denoted as \((3, D, 4)\)-graphs. We obtain all non-isomorphic \((3, D, 4)\)-graphs for \( D = 2 \). Furthermore, for any diameter \( D \), we consider the girth of \((3, D, 4)\)-graphs. By a counting argument, it is easy to see that the girth is at least \( 2D – 2 \). The main contribution of this paper is that we prove that the girth of a \((3, D, 4)\)-graph is at least \( 2D – 1 \). Finally, for \( D > 4 \), we conjecture that the girth of a \((3, D, 4)\)-graph is \( 2D \).

A fast direct method for obtaining the incidence matrix of a finite projective plane of order \( n \) via \( n-1 \) mutually orthogonal \( n \times n \) Latin squares is described. Conversely, \( n-1 \) mutually orthogonal \( n \times n \) Latin squares are directly exhibited from the incidence matrix of a projective plane of order \( n \). A projective plane of order \( n \) can also be described via a digraph complete set of Latin squares, and a new procedure for doing this will also be described.

The Fibonacci graph \( G_n \) is the graph whose vertex set is the collection of \( n \)-bit binary strings having no contiguous ones, and two vertices are adjacent if and only if their Hamming distance is one. Values of several graphical invariants are determined for these graphs, and bounds are found for other invariants.

Given a configuration of pebbles on the vertices of a connected graph \( G \), a \({pebbling\; move}\) is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. We introduce the notion of domination cover pebbling, obtained by combining graph cover pebbling with the theory of domination in graphs. The domination cover pebbling number, \( \psi(G) \), of a graph \( G \) is the minimum number of pebbles that are placed on \( V(G) \) such that after a sequence of pebbling moves, the set of vertices with pebbles forms a dominating set of \( G \), regardless of the initial configuration of pebbles. We discuss basic results and determine \( \psi(G) \) for paths, cycles, and complete binary trees.

We show that the double domination number of an \( n \)-vertex, isolate-free graph with minimum degree \( \delta \) is bounded above by \(\frac{n(\ln(\delta + 1) + \ln \delta + 1)}{\delta}.\) This result improves a previous bound obtained by J. Harant and M. A. Henning [On double domination in graphs, \({Discuss. Math. Graph Theory}\) \({25} (2005), 29-34]\). Further, we show that for fixed \( k \) and large \( \delta \), the \( k \)-tuple domination number is at most \(\frac{n(\ln \delta + (k – 1 + o(1))\ln \ln \delta)}{\delta},\) a bound that is essentially best possible.

Let \(\alpha\)-resolvable STS(\(v\)) denote a Steiner triple system of order \(v\) whose blocks are partitioned into classes such that each point of the design occurs in precisely \(\alpha\) blocks in each class. We show that for \(v \equiv u \equiv 1 \pmod{6}\) and \(v \geq 3u + 4\), there exists an \(\alpha\)-resolvable STS(\(v\)) containing an \(\alpha\)-resolvable sub-STS(\(u\)) for all suitable \(\alpha\).

A vertex set \( S \subseteq V(G) \) of a graph \( G \) is a \( 2 \)-dominating set of \( G \) if \( |N(v) \cap S| \geq 2 \) for every vertex \( v \in (V(G) – S) \), where \( N(v) \) is the neighborhood of \( v \). The \( 2 \)-domination number \( \gamma_2(G) \) of graph \( G \) is the minimum cardinality among the \( 2 \)-dominating sets of \( G \). In this paper, we present the following Nordhaus-Gaddum-type result for the \( 2 \)-domination number. If \( G \) is a graph of order \( n \), and \( \bar{G} \) is the complement of \( G \), then

\[ \gamma_2(G) + \gamma_2(\bar{G}) \leq n + 2, \]

and this bound is best possible in some sense.

The Graph Isomorphism (GI) problem asks if two graphs are isomorphic. Algorithms which solve GI have applications in, but not limited to, SAT solver engines, isomorph-free generation, combinatorial analysis, and analyzing chemical structures. However, no algorithm has been found which solves GI in polynomial time, implying that hard instances should exist. One of the most popular algorithms, implemented in the software package nauty, canonically labels a graph and outputs generators for its automorphism group. In this paper, we present some methods that improve its performance on graphs that are known to pose difficulty.

Let \( C \) be the set of distinct ways in which the vertices of a \( 5 \)-cycle may be coloured with at most two colours, called \({colouring\; types}\), and let \( S \subseteq C \). Suppose we colour the vertices of \( K_v \) with at most two colours. If \( \mathcal{D} \) is a \( 5 \)-cycle decomposition of \( K_v \), such that the colouring type of each \( 5 \)-cycle is in \( S \), and every colouring type in \( S \) is represented in \( \mathcal{D} \), then \( \mathcal{D} \) is said to have a \emph{proper colouring type} \( S \). For all \( S \) with \( |S| \leq 2 \), we determine some necessary conditions for the existence of a \( 5 \)-cycle decomposition of \( K_v \) with proper colouring type \( S \). In many cases, we show that these conditions are also sufficient.

Most computer algebra packages for Weyl groups use generators and relations and the Weyl group elements are expressed as reduced words in the generators. This representation is not unique and leads to computational problems. In [HHR06], the authors introduce the representation of Weyl group elements uniquely as signed permutations. This representation is useful for the study of symmetric spaces and their representations.

A computer algebra package enabling one to do computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this paper, we use the representation of Weyl group elements as signed permutations to improve the algorithms of [DH05]. These algorithms compute the fine structure of symmetric spaces and nice bases for local symmetric spaces.