For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...

‎The first variable Zagreb index of graph $G$ is defined as‎ ‎begin{eqnarray*}‎ ‎M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}‎, ‎end{eqnarray*}‎ ‎where $lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (rec...

We give a formula that expresses a sum of products of poly-Bernoulli numbers of negative index as a linear combination of poly-Bernoulli numbers. More generally, we show that if a two-variable formal power series satisfies a certain partial differential equation, then its coefficients satisfy this type of formula. As an appendix, we solve this partial differential equation.

The stress of a vertex is node centrality index, which has been introduced by Shimbel (1953). in graph the number geodesics (shortest paths) passing through it. In this paper, we introduce new topological index for graphs called square root sum using stresses vertices. Further, establish some inequalities, prove results and compute stress-sum standard graphs.

OBJECTIVE We seek to develop a new approach for analyzing the physical demands of highly variable lifting tasks through an adaptation of the Revised NIOSH (National Institute for Occupational Safety and Health) Lifting Equation (RNLE) into a Variable Lifting Index (VLI). BACKGROUND There are many jobs that contain individual lifts that vary from lift to lift due to the task requirements. The ...