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).
