Open Journal of Discrete Applied Mathematics (ODAM)

The Wiener index \(W(G)\) of a graph \(G\) is defined as the sum of distances between its vertices. A tree \(T\) generates \(r\)-uniform hypergraph \(H_{r,k}(T)\) by the following way: hyperedges of cardinality \(r\) correspond to edges of the tree and adjacent hyperedges have \(k\) vertices in common. A relation between quantities \(W(T)\) and \(W(H_{r,k}(T))\) is established.

The aim of this paper is to calculate the multiplicative topological indices of Zigzag polyhex nanotubes, Armchair polyhex nanotubes, Carbon nanocone networks, two dimensional Silicate network, Chain silicate network, six dimensional Hexagonal network, five dimensional Oxide network and four dimensional Honeycomb network.

Let \(G\) be a simple connected graph with \(n\) vertices, \(m\) edges, and a sequence of vertex degrees \(\Delta=d_1\geq d_2\geq\cdots\geq d_n=\delta >0\). Denote by \(\mu_1\geq \mu_2\geq\cdots\geq \mu_{n-1}>\mu_n=0\) the Laplacian eigenvalues of \(G\). The Kirchhoff index of \(G\) is defined as \(Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}\). A couple of new lower bounds for \(Kf(G)\) that depend on \(n\), \(m\), \(\Delta\) and some other graph invariants are obtained.

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.