in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

The asymptotic distributions of the number of vertices of given degree in random graph K n,p are given. By using the method of Poisson convergence, Poisson and normal distributions are obtained.

Upon the discovery of power laws, a large body of work in complex network analysis has focused on developing generative models of graphs which mimick real-world network properties such as skewed degree distributions, small diameter and large clustering coefficients. Despite the fact that planar graphs arise in numerous real-world settings, e.g., in road and railway maps, in printed circuits etc...

Let T be a tree and n_{l}(eIT) and n_{2}(eIT) denote the number of vertices of T, lying on the two sides of the edge e. Suppose T_{l} and T_{2} are two trees with equal number of vertices, e in T_{1} and f in T_{2}. The edges e and f are said to be equiseparable if either n_{l}(eIT_{I}) = n_{l}(fIT_{2}) or n_{l}(eIT_{I}) = n_{2}(fIT_{2}). If there is an one-to-one correspondence between the ver...

Let g(n, m) denote the class of simple graphs on n vertices and m edges and let C E 9( a, m). For suitably restricted values of m, C will necessarily contain certain prescribed subgraphs such as cycles of given lengths and complete graphs . For example> if m > 7nZ then G contains cycles of all lengths up to Li(n+ 3) J . Recently we have established a number of results concerning the existence o...