Open Journal of Discrete Applied Mathematics (ODAM)

In this note, we first show that the general Zagreb index can be obtained from the \(M-\)polynomial of a graph by giving a suitable operator. Next, we obtain \(M-\)polynomial of some cactus chains. Furthermore, we derive some degree based topological indices of cactus chains from their \(M-\)polynomial.

Let \(G= G(V,E)\) be a \((p,q)\)-graph. A bijection \(f: E\to\{1,2,3,\ldots,q \}\) is called an edge-prime labeling if for each edge \(uv\) in \(E\), we have \(GCD(f^+(u),f^+(v))=1\) where \(f^+(u) = \sum_{uw\in E} f(uw)\). A graph that admits an edge-prime labeling is called an edge-prime graph. In this paper we obtained some sufficient conditions for graphs with regular component(s) to admit or not admit an edge-prime labeling. Consequently, we proved that if \(G\) is a cubic graph with every component is of order \(4, 6\) or \(8\), then \(G\) is edge-prime if and only if \(G\not\cong K_4\) or \(nK(3,3)\), \(n\equiv2,3\pmod{4}\). We conjectured that a connected cubic graph \(G\) is not edge-prime if and only if \(G\cong K_4\).

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.