Sharp bounds and normalization of Wiener-type indices

10.1371/journal.pone.0078448PLoS ONE811-POLN

Nano-Zagreb Index and Multiplicative Nano-Zagreb Index of Some Graph Operations

Let G be a graph with vertex set V(G) and edge set E(G). The Nano-Zagreb and multiplicative Nano-Zagreb indices of G are NZ(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)) and N*Z(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)), respectively, where d(v) is the degree of the vertex v. In this paper, we define two types of Zagreb indices based on degrees of vertices. Also the Nano-Zagreb index and multiplicative Nano-Zagreb index of the Cartesian product, symmetric difference, composition and disjunction of graphs are computed

New transmission irregular chemical graphs

The transmission of a vertex $v$ of a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is a transmission irregular graph. It is shown that for any odd number $n\geq 7$ there exists a transmission irregular chemical tree of order $n$. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge