Edge-Nim is a combinatorial game played on finite regular graphs with positive, integrally weighted edges. Two players alternately move from an initialized vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly non-negative integer as they travel across it. The game ends when no incident edge has a nonzero weight and a player is unable to move, in which case that player loses.

We characterize the winner of Edge-Nim on the complete bipartite graphs \( K_{2,n} \) for all positive integers \( n \), giving the solution and complete strategy for the player able to win.

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem.

In this paper, we investigate the pansophy of two classes of graphs that contain a vertex that we define as the superuser. The superuser of a graph is a vertex that is adjacent to every other vertex. We close with future research directions.

A grid on a cell of a game board attacks all neighboring cells. The domination number counts the minimum number of grids such that each cell of a board is occupied or attacked by a grid.

For square boards (chess boards), the domination number has been determined in a series of papers. Here, we start to consider grids on hexagon boards \( B_n \) as parts of the Euclidean tessellation by congruent regular hexagons, where \( B_1 \) is one hexagon, \( B_2 \) consists of the three hexagons around one vertex, and \( B_n \) for \( n \geq 3 \) consists of \( B_{n-2} \) together with all hexagons having at least one hexagon in common with \( B_{n-2} \).

An upper bound is presented for the grid domination number, and exact values are determined by computer for small \( n \).

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem.

In this paper, we construct a method to determine the pansophy of any complete bipartite graph, and then generalize the method to compute the pansophy of any complete multipartite graph. We close with future research directions.

The \emph{Reconstruction Number} of a graph \( G \), denoted \( RN(G) \), is the minimum number \( k \) such that there exist \( k \) vertex-deleted subgraphs of \( G \) which determine \( G \) up to isomorphism. More precisely, \( RN(G) = k \) if and only if there are vertex-deleted subgraphs \( G_1, G_2, \ldots, G_k \), such that if \( H \) is any graph with vertex-deleted subgraphs \( H_1, H_2, \ldots, H_k \), and \( G_i \cong H_i \) for \( i = 1, 2, \ldots, k \), then \( G \cong H \).

A \emph{unicyclic graph} is a connected graph with exactly one cycle. In this paper, we find reconstruction numbers for various types of unicyclic graphs. With one exception, all unicyclic graphs considered have \( RN(G) = 3 \).

In this work, we introduce the Interval Permutation Segment (IP-SEG) model that naturally generalizes the geometric intersection models of interval and permutation graphs.
We study properties of two graph classes that arise from the IP-SEG model and present a family of forbidden subgraphs for these classes. In addition, we present polynomial algorithms for the following problems on these classes, when the model is given as part of the input.

Let \( S_n \) be the random walk on \( (0, 1) \). The \( S_n \) have been the subject of intense study; their definition is immediately intuitive. Nevertheless, they are quite intentionally disorderly, and this disorder is mirrored by the fact that, pointwise, \( \left( \frac{S_n}{\sqrt{n}} \mid n \in \mathbb{N}^+ \right) \) behaves quite badly.

In this paper, we provide our results on the fine structure of the random walk that give insight into this behavior.

We utilize the flexible tile model presented in [13] to design self-assembling DNA structures from a graph theory perspective. These tiles represent branched junction molecules whose arms are double strands of DNA.

We consider \( 2 \times n \) triangular lattice graphs \( G_n \), where \( n \) represents the number of triangles. Given a target graph \( G_n \), we determine the minimum number of tile and bond-edge types needed in order to create \( G_n \) as a complete self-assembled complex in three different scenarios. Each scenario corresponds to a distinct level of laboratory constraint.

In the first scenario, graphs of a smaller size than \( G_n \) are allowed. In the second scenario, non-isomorphic graphs of the same size as \( G_n \) are allowed, but not graphs of smaller size. In the third scenario, only graphs isomorphic or larger in size to the target graph are allowed.

We provide optimal tile sets for all \( 2 \times n \) triangular lattice graphs \( G_n \) in Scenario 1 and Scenario 3. We also include some small examples in Scenario 2.

An upper bound on the energy of graphs is obtained using the spectral moments of the eigenvalues of the adjacency matrix associated with the graph, utilizing the method of Lagrange multipliers and properties of cubic equations

A polyhex is a set of hexagons of the Euclidean tessellation of the plane by congruent regular hexagons. Then, a polyhex graph has the vertex points of the hexagons as its vertices and the sides of the hexagons as its edges. A rectilinear drawing of a graph in the plane uses straight line segments for the edges. Partial results are given for the maximum number of crossings over all rectilinear drawings of a polyhex graph