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

Distinctive power of the alliance polynomial has been studied in previous works. For instance, it has been proved that the empty, path, cycle, complete, complete without one edge, and star graphs are characterized by its alliance polynomial. Moreover, it has been proved that the family of alliance polynomials of regular graphs with small degree is a very
special one, since it does not contain alliance polynomials of graphs other than regular graphs with the same degree. In this work, we prove that the alliance polynomial also
determines the wheel graphs.

An ordered tree, also known as a plane tree or a planar tree, is defined recursively as having a root and an ordered set of subtrees. A \(3\)-zebra tree is an ordered tree where all edges connected to the root (called height \( 1 \)) are tricolored, as are all edges at odd height. The edges at even height are all black as usual.

In this paper, we show that the number of \(3\)-zebra trees with \( n \) edges is equal to the number of Schröder paths with bicolored level steps.

A split graph is a graph whose vertices can be partitioned into a clique and an independent set. Most results in spectral graph theory do not address multigraph concerns. Exceptions are [2] and [4], but these papers present results involving a special class of underlying split graphs, threshold graphs, in which all pairs of nodes exhibit neighborhood nesting, and all multiple edges are confined to the clique.

We present formulas for the eigenvalues of some infinite families of regular split multigraphs in which all multiple edges occur between the clique nodes and cone nodes, with multiplicity of multiple edges \( \mu > 1 \) fixed, and which have integer eigenvalues for the adjacency, Laplacian, and signless Laplacian matrices.

A rigid vertex is a vertex with a prescribed cyclic order of its incident edges. An embedding of a rigid vertex graph preserves such a cyclic order in the surface at every vertex. A cellular embedding of a graph has the complementary regions homeomorphic to open disks.

The genus range of a \( 4 \)-regular rigid vertex graph \( \Gamma \) is the set of genera of closed surfaces that \( \Gamma \) can be cellularly embedded into. Inspired by models of DNA rearrangements, we study the change in the genus range of a graph \( \Gamma \) after the insertion of subgraph structures that correspond to intertwining two edges. We show that such insertions can increase the genus at most by \( 2 \) and decrease by at most \( 1 \), regardless of the number of new vertices inserted.