Open Journal of Discrete Applied Mathematics (ODAM)

Zaslavsky (1991) characterized all single-element coextensions of graphic matroids in terms of a graphical structure called a biased graph. In this paper we characterize all orientations of a single-element coextension of a graphic matroid in terms of graphically defined orientations of its associated biased graph.

The exact deg-centric graph of a simple graph \(G\), denoted by \(G_{ed}\), is a graph constructed from \(G\) such that \(V(G_{ed}) = V(G)\) and \(E(G_{ed}) = \{v_iv_j: d_G(v_i,v_j) = deg_G(v_i)\}\). This research note presents the domination numbers of both the Jaco graph \(J_n(x)\) and the exact deg-centric graph of the family of Jaco graphs. The respective complement graphs are also addressed.

The first Zagreb index of a graph is one of the most important topological indices in chemical graph theory. It is also an important invariant of general graphs. The first Zagreb index of a graph is defined as the sum of the squares of the degrees of the vertices in the graph. The research on the Hamiltonian properties of a graph is an important topic in graph theory. Use the Diaz-Metcalf inequality, we in this paper present new sufficient conditions based on the first Zagreb index for the Hamiltonian and traceable graphs. In addition, using the ideas of obtaining the sufficient conditions, we also present an upper bound for the first Zagreb index of a graph. The graphs achieving the upper bound are also characterized.

This paper introduces the concept of the extended \(H\)-cover of a graph \(G\), denoted as \(G^*_H\) , as a generalization inspired by the extended double cover graphs discussed in Chen [1]. We explore the spectral properties and energy characteristics of \(G^*_H\), deriving formulae for the number of spanning trees in cases where both \(G\) and \(H\) are regular. Our investigation identifies several infinite families of equienergetic graphs and highlights instances of cospectral graphs within \(G^*_H\) . Additionally, we analyze various graph parameters related to the Indu-Bala product of graphs and the partial complement of the subdivision graph (PCSD) of \(G\).

A dominator coloring of a graph \(\mathscr{G}\) is a proper coloring where each vertex of \(\mathscr{G}\) is within the closed neighborhood of at least one vertex from each color class. The minimum number of color classes required for a dominator coloring of \(\mathscr{G}\) is termed the dominator chromatic number. Additionally, a total dominator coloring of a graph \(\mathscr{G}\) is a proper coloring in which every vertex dominates at least one color class other than its own. The minimum number of color classes needed for a total dominator coloring of \(\mathscr{G}\) is known as the total dominator chromatic number. In this paper, our objective is to derive findings concerning dominator and total dominator coloring of the duplication corresponding corona of specific graphs.

The topological index is a molecular property that is determined from a chemical compound’s molecular graph. Topological indices are numerical graph parameters that inform us about the topology of the graph and are generally graph invariants. In this paper, we consider some topological indices based on the second distance of each vertex of the graph \(\alpha\) and the number of unordered pairs of vertices \(\{s,q\} \subseteq V(\alpha)\) which are at distance \(3\) in \(\alpha\). These indices are called the leap Zagreb index and the Wiener polarity index, respectively. we compute these indices of \(R\)-vertex join and \(R\)-edge join of graphs.

A novel topological index, the Sombor index, has been proposed by Ivan Gutman in a recent paper [1]. Motivated by this novel index, we study the new variants of Sombor index and to examine the correlation of newly introduced topological indices we have computed the values of these indices by taking all possible trees on 10 vertices. Here in this paper, we derive explicit formulae for the Sombor index of various nanostructures. These include hexagonal parallelogram \( P(\alpha, \beta) \)-nanotubes, triangular benzenoid \( G_{\alpha} \), and zigzag-edge coronoid fused with starphene nanotubes \( ZCS(k,\alpha,\beta) \), where \( k, \alpha, \beta \) are natural numbers. We also compute the Sombor index for dominating derived networks \( D_{1}, D_{2}, D_{3} \), as well as for various dendrimers such as Porphyrin Dendrimer, Ninc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimers, and Polyamidoamin (PAMAM) Dendrimer. Additionally, we examine Polyamidoamin dendrimers (\( PD_{1}, PD_{2}, DS_{1} \)) and linear polyomino chains like \( L_{\alpha} \), \( Z_{\alpha} \), \( B^{1}_{\alpha}(\alpha \geq 3) \), \( B^{2}_{\alpha}(\alpha \geq 4) \). Finally, we consider benzenoid systems with different shapes, including triangular, hourglass, and jagged-rectangle configurations. By computing the Sombor index for these nanostructures, we provide a comprehensive analysis of their topological properties.

The hub set measures the connectivity of any nodes in graphs and the determination of it is found to be NP-complete. This paper deduces several properties and characterize of one such hub parameter, the doubly connected hub number for its value equal to 1 and 2. Moreover, a few bounds and Nordhaus-Gaddum type inequalities are discussed.

Let \( V(G) = \{v_1, v_2, \ldots, v_n\} \) be the vertex set and \( E(G) = \{e_1, e_2, \ldots, e_m\} \) be the edge set of a graph \( G \). The Seidel adjacency matrix of a graph \( G \) is defined as \( S(G) = [s_{ij}] \) of order \( n \times n \), in which \( s_{ij} = -1 \) if \( v_i \) is adjacent to \( v_j \), \( s_{ij} = 1 \) if \( v_i \) is not adjacent to \( v_j \) and \( s_{ii} = 0 \). We introduce here the \( (-1,1) \)-incidence matrix of \( G \) as \( B_S(G) = [c_{ij}] \) of order \( n \times m \), in which \( c_{ij} = -1 \) if \( v_i \) is incident to \( e_j \) and \( c_{ij} = 1 \) if \( v_i \) is not incident to \( e_j \). Further we explore properties of \( B_S(G) \) and of its transpose.

Faces in graphs play a crucial role in understanding the structural properties of planar graphs. They represent the regions or bounded areas formed by the edges of the graph when it is embedded in the plane. The concept of faces provides insights into the connectivity and layout of systems, helping analyze the geometry and topology of networks, communication systems, and various real-world applications. In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances. In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances. Let \( G \) be a connected planar graph with vertex \( V(G) \), edge set \( E(G) \), and face set \( F(G) \). The distance between a face \( f \) and a vertex \( v \) is defined as the minimum distance from \( v \) to any vertex incidence to \( f \). In this work, we introduce a new resolvability parameter for connected planar graphs, referred to as the face metric dimension. A face-resolving set \( R \subseteq V(G) \) is a set of vertices such that for every pair of distinct faces \( f_1, f_2 \in F(G) \), there exists at least one vertex \( r \in R \) for which the distances \( d(f_1, r) \) and \( d(f_2, r) \) are distinct. The face metric dimension of \( G \), denoted \( \ fmd(G) \), is the minimum cardinality of a face-resolving set. This new metric provides insight into the structure of planar graphs and offers a novel perspective on the analysis of graph resolvability.