Open Journal of Discrete Applied Mathematics (ODAM)

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.

A finite, connected simple graph \(G\) is a geodetic graph if and only if for each pair of vertices \(v_i, v_j\) there exists a unique distance path (or unique shortest \(v_iv_j\)-path). The insertion of vertices in an edge or edges of a non-geodetic graph \(G\) to, if possible, obtain a resultant geodetic graph is called geodetication of the graph \(G\). The paper introduces two new graph parameters generally called the Ruv\(\acute{e}\) numbers of a graph. The Ruv\(\acute{e}\) numbers of \(G\) are denoted by \(\rho_1(G)\) and \(\rho_2(G)\) respectively, and \(\rho_1(G) = \rho_2(G) = 0\) if and only if \(G\) is geodetic. Furthermore, for some graphs the parameter, \(\rho_1(G) \to \infty\). The latter graphs \(G\) do not permit geodetication in respect of \(\rho_1(G)\). It is evident that geodetication presents various challenging minimization problems. The core field of application will be, restricting graphs to distance path uniqueness. Intuitive applications are foreseen in military science, IT anti-hacking coding and predictive flow through networks.

The anchored Dyck words of length \(n=2k+1\) (obtained by prefixing a 0-bit to each Dyck word of length \(2k\) and used to reinterpret the Hamilton cycles in the odd graph \(O_k\) and the middle-levels graph \(M_k\) found by M\”utze et al.) represent in \(O_k\) (resp., \(M_k\)) the cycles of an \(n\)- (resp., \(2n\)-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS’s), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming \(i\)-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on \(k\).

In this paper, we introduce a new resolvability parameter named as the local edge partition dimension \((LEPD)\) of graphs. The local edge partition dimension \((LEPD)\) makes a specialty of partitioning the vertex set of a graph into awesome instructions based totally on localized resolving properties. Our findings offer a fresh angle on graph resolvability, offering capability insights for optimizing network overall performance and structural analysis. Let \(G=(V, E)\) be a connected graph with vertex set \(V\) and edge set \(E\). A partition set \({R}_{p}=\{{R}_{p1},{R}_{p2},{R}_{p3}\dots,{R}_{pn}\}\) contain subsets of vertices of \(G\). If for every pair of adjacent edges \(p\) and \(q\) in \(G\), then \(d(p,{R}_{p})\neq d(q,{R}_{p})\) and if \(p\) and \(q\) are non-adjacent then not necessary \(d(p,{R}_{p})\neq d(q,{R}_{p})\) then \({R}_{p}\) is called a local edge resolving partition set and minimum cardinality of such set is called local edge partition dimension. We discussed local metric, local edge metric, metric, edge metric dimension, local partition, local edge partition, partition dimension, and edge partition dimension of the Petersen graph.

In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by converting a collection of numbers encoding certain combinatorial numerics from objects exhibiting symmetry in various dimensions. Using the algorithms first described in [1] we show that the NIST testing suite described in publication 800-22 does not detect these symmetries hidden within these sequences.

This note addresses impracticalities or possible absurdities with regards to the definition corresponding of some graph parameters. To remedy the impracticalities the principle of transmitting the definition is put forward. The latter principle justifies a comprehensive review of many known graph parameters, the results related thereto, as well as the methodology of applications which draw a distinction between connected versus disconnected simple graphs. To illustrate the notion of transmitting the definition, various parameters are re-examined such as, connected domination number, graph diameter, girth, vertex-cut, edge-cut, chromatic number, irregularity index and quite extensively, the hub number of a graph. Ideas around undefined viz-a-viz permissibility viz-a-viz non-permissibility are also discussed.