There exist many topological indices that are calculated on saturated hydrocarbons since they can be easily modelled by simple graphs. On the other hand, it is more challenging to investigate topological indices for hydrocarbons with multiple bonds. The purpose of this paper is to introduce a simple model that gives good results for predicting physico-chemical properties of alkenes and alkadienes. In particular, we are interested in predicting boiling points of these molecules by using the well known Wiener index and its weighted versions. By performing the non-linear regression analysis we predict boiling points of alkenes and alkadienes.

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs $P(ck,k)$. Exact values are established for all infinite families where the general lower bound $\frac{4}{5}ck$ is attained. In all other cases lower and upper bounds with small gaps are given.

A k-rainbow dominating function (kRDF) f of G assigns subsets of \(\{1,2,\ldots ,k\}\) to vertices, such that for vertex v with \(f(v)=\emptyset\), \(\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}\). The weight w(f) of kRDF f is \(w(f)=\sum _{v\in V(G)}\left| f(v)\right|\). The minimum weight of a kRDF of G is the k-rainbow domination number denoted by \(\gamma _{rk}(G)\). This paper focuses on the 2-rainbow domination number of Cartesian graph bundles of cycles over cycles, extending recent results for Cartesian product of cycles. Exact values are given for certain infinite families, and tight lower and upper bounds are established for general case.

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss numerical results. As polynomials, we use, e.g., the Hosoya, the Schultz, and the Gutman polynomial which belong to an interesting family of degree-distance-based graph polynomials; they constitute so-called counting polynomials with non-negative integers as coefficients and the roots of their modified versions have been used to characterize the topology of graphs. Our results can be applied for the quantitative characterization of graphs. Besides analytical results on bounds and convergence, we also investigate other properties of those measures such as their degeneracy which is an undesired aspect of graph measures. It turns out that the measures representing roots of graph polynomials possess high discrimination power on exhaustively generated trees, which outperforms standard versions of these indices. Furthermore, a new measure is introduced that allows us to compare different topological indices in terms of structure sensitivity and abruptness.

A Roman dominating function f of a graph $G=(V,E)$ assigns labels from the set $\{0,1,2\}$ to vertices such that every vertex labeled 0 has a neighbor labeled 2. The weight of an RDF f is defined as $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$, and the Roman domination number, ${\gamma }_R\left(G\right)$, is the minimum weight among all RDFs of G. This paper studies the domination and Roman domination numbers in Cartesian bundles of cycles. Furthermore, the constructed optimal patterns improve known bounds and suggest even better bounds might be achieved by combining patterns, especially for bundles involving shifts of order $4k$ and $5k$.