Let \( G = (V, E) \) be a finite graph, where \( V(G) \) and \( E(G) \) are the (non-empty) sets of vertices and edges of \( G \). An \((a, d)\)-\({edge-antimagic\; total\; labeling}\) is a bijection \( \beta \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) with the property that the set of all the edge-weights, \( w(uv) = \beta(u) + \beta(uv) + \beta(v) \), for \( uv \in E(G) \), is \( \{a, a + d, a + 2d, \dots, a + (|E(G)| – 1)d\} \), for two fixed integers \( a > 0 \) and \( d \geq 0 \). Such a labeling is super if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings for disjoint unions of multiple copies of a regular caterpillar.

The term mode graph was introduced by Boland, Kaufman, and Panrong to define a connected graph \( G \) such that, for every pair of vertices \( v, w \) in \( G \), the number of vertices with eccentricity \( e(v) \) is equal to the number of vertices with eccentricity \( e(w) \). As a natural extension to this work, the concept of an antimode graph was introduced to describe a graph for which, if \( e(v) \neq e(w) \), then the number of vertices with eccentricity \( e(v) \) is not equal to the number of vertices with eccentricity \( e(w) \). In this paper, we determine the existence of some classes of antimode graphs, namely equisequential and \((a, d)\)-antimode graphs.

By an \((a, d)\)-edge-antimagic total labeling of a graph \( G(V, E) \), we mean a bijective function \( f \) from \( V(G) \cup E(G) \) onto the set \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) such that the set of all the edge-weights, \( w(uv) = f(u) + f(uv) + f(v) \), for \( uv \in E(G) \), is \( \{a, a+d, a+2d, \dots, a + (|E(G)| – 1)d\} \), for two integers \( a > 0 \) and \( d \geq 0 \).

In this paper, we study the edge-antimagic properties for the disjoint union of complete \( s \)-partite graphs.

We study the number of super edge-magic (bipartite) graphs from an asymptotic point of view.

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.