Open Journal of Discrete Applied Mathematics (ODAM)

In this article, we study some properties of the solutions of the following difference equation: \(x_{n+1}=a x_{n}+\dfrac{b x_{n} x_{n-4}}{cx_{n-3}+dx_{n-4}},\quad n=0,1,…\) where the initial conditions \(x_{-4},x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary positive real numbers and \(a, b, c, d\) are positive constants. Also, we give specific form of the solutions of four special cases of this equation.

For a given connected graph \(G\) and a real number \(\alpha\), denote by \(d(u)\) the degree of vertex \(u\) of \(G\), and denote by \(\chi_{\alpha}(G)=\sum_{uv\in E(G)} \big(d(u)+d(v)\big)^{\alpha}\) the general sum-connectivity index of \(G\). In the present note, we determine the smallest general sum-connectivity index of trees (resp., chemical trees) together with corresponding extremal trees among all trees (resp., chemical trees) with \(n\) vertices and \(k\) pendant vertices for \(0<\alpha<1.\)

An outer-connected vertex edge dominating set (OCVEDS) for an arbitrary graph \(G\) is a set \(D \subset V(G)\) such that \(D\) is a vertex edge dominating set and the graph \(G \setminus D\) is connected. The outer-connected vertex edge domination number of \(G\) is the cardinality of a minimum OCVEDS of \(G\), denoted by \(\gamma_{ve}^{oc}(G)\). In this paper, we give the outer-connected vertex edge dominating set in lexicographic product of graphs.

The minimum degree matrix \(MD(G)\) of a graph \(G\) of order \(n\) is an \(n\times n\) symmetric matrix whose \((i,j)^{th}\) entry is \(min\{d_i,d_j\}\) whenever \(i\neq j,\) and zero otherwise, where \(d_i\) and \(d_j\) are the degrees of the \(i^{th}\) and \(j^{th}\) vertices of \(G\), respectively. In the present work, we obtain the minimum degree polynomial of the graphs obtained by some graph operators (generalized \(xyz\)-point-line transformation graphs).

The exact solutions of most nonlinear difference equations cannot be obtained theoretically sometimes. Therefore, a massive number of researchers predict the long behaviour of most difference equations by investigating some qualitative behaviours of these equations from the governing equations. In this article, we aim to analyze the asymptotic stability, global stability, periodicity of the solution of an eighth-order difference equation. Moreover, a theoretical solution of a special case equation will be presented in this paper.

Topological indices are real numbers associated with molecular graphs of compounds that help to guess properties of compounds. Hex-Derived networks has an assortment of valuable applications in drug store, hardware, and systems administration. Imran et al. [1] computed the general Randić, first Zagreb, ABC, GA, ABC\(_{4}\), and GA\(_{5}\) indices for these hex-derived networks. In this article, we extend the work of [1] and compute some new topological indices of these networks.

The mathematical chemistry deals with applications of graph theory to study the physicochemical properties of molecules theoretically. A chemical graph is a simple graph where hydrogen depleted atoms are vertices and covalent bonds between them represent the edges. A topological index of a graph is a numeric quantity obtained from the graph mathematically. A cactus graph is a connected graph in which no edges lie in more than one cycle. In this study, we derive exact expressions of general Zagreb index of some cactus chains.

In this paper, the covering radius of codes over \(\mathbb R ={\mathbb Z_2}{R^{*}},\) where \(R^{*}={\mathbb Z_2}+v{\mathbb Z_2},v^{2}=v\) with different weight are discussed. The block repetition codes over \(\mathbb R\) is defined and the covering radius for block repetition codes, simplex code of \(\alpha\)-type and \(\beta\)-type in \(\mathbb R\) are obtained.

The atom-bond connectivity (ABC) index of a graph \(G=(V,E)\) is defined as \(ABC(G)=\sum_{v _{i}v_{j} \in E}\sqrt{(d_{i}+d_{j}-2)/(d_{i}d_{j})}\), where \(d_{i}\) denotes the degree of vertex \(v_{i}\) of \(G\). Due to its interesting applications in chemistry, this molecular structure descriptor has become one of the most actively studied vertex-degree-based graph invariants. Many efforts were made towards the elementary problem of characterizing tree(s) with minimal ABC index, which remains open and was coined as the ABC index conundrum”. Up to date, quite a few significant results have been obtained. In the course of research computer search plays a non-negligible role. In the present paper we review the state of the art of the problem. In addition we intend to demonstrate that, repeating the procedure “searching – conjecturing – proving” can be an applicable paradigm to cope with elusive problems of extremal graph characterization.

Let \(G\) be a simple, finite and connected graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). Assume that \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\) and if each (H_i\), \(1 \leq i \leq k\), is a path or cycle in \(G\), then the collection of edge-disjoint subgraphs of \(G\) denoted by \(\psi\) is called a path decomposition of \(G\). If each \(H_i\) is a path in \(G\) then \(\psi\) is called an acyclic path decomposition of \(G\). The minimum cardinality of a path decomposition of \(G\), denoted by \(\pi (G)\), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of \(G\), denoted by \(\pi_a(G)\), is called the acyclic path decomposition number of \(G\). In this paper, we determine path decomposition number for a number of graphs in particular, the Cartesian product of graphs. We also provided bounds for \(\pi(G)\) and \(\pi_a(G)\) for these graphs.