Irregularity indices are generally used for quantitative characterization of topological structure of non-regular graphs. According to a widely accepted preconception, using a topological invariant (called a graph irregularity index) for that purpose, the results of graph irregularity classification should be consistent with our subjective judgements (intuitive feeling). In the case of structurally strongly similar graphs, it is difficult to select the proper irregularity index by which the irregularity ranking (ordering) of graphs can be per- formed in a consistent manner to our preliminary expectations. In this work we investigate various possibilities of constructing so-called composite irregularity indices obtained as a function of traditional irregularity indices and test their discriminatory performance. More- over, it has been demonstrated in examples, that in some cases the subjective evaluation of graph irregularities leads to false conclusions. This phenomenon is called the paradox of quantitative graph irregularity characterization. From our study we have concluded that results of graph irregularity measuring depend not only on the choice of irregularity indices but it is inuenced strongly on the preselected set of graphs to be investigated. Similar problems arise for the quantitative evaluation of information content, complexity or branching of molecular graphs.

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set $\{x_1,\dots ,x_n\}$, then the ad-hoc Lanzhou index of G is defined by $\widetilde{Lz}\left(G\right)={\sum }_{i=1}^n{d}_i{(n-1-d_i)}^2$, where $d_i$ represents the degree of the vertex $x_i$. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected $\xi$-cyclic graphs of a fixed order, (ii) the set of all connected molecular $\xi$-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

An irregularity index $IR\left(\mathsf{\Gamma }\right)$ of a graph $\mathsf{\Gamma }$ is a nonnegative numeric quantity (i.e., $IR\left(\mathsf{\Gamma }\right)\ge 0$) such that $IR\left(\mathsf{\Gamma }\right)=0$ iff $\mathsf{\Gamma }$ is a regular graph. In this paper, we show that $IRC$ closely correlates with the normal boiling point $T_{bp}$ and the standard heat of formation $\Delta H_f^o$ of lower benzenoid hydrocarbons. The correlation models that fit the data efficiently for both $T_{bp}$ and $\Delta H_f^o$ are linear. We develop further mathematical properties of $IRC$ by calculating its exact expressions for the recently introduced transformation graphs as well as certain derived graphs, such as the total graph, semi-total point graph, subdivision graph, semi-total line graph, double, strong double, and extended double cover graphs. Some open problems are proposed for further research on the $IRC$ index of graphs.

Let G be a connected (molecular) graph with the vertex set $V\left(G\right)=\{v_1,\cdots ,v_n\}$, and let $d_i$ and ${\sigma }_i$ denote, respectively, the vertex degree and the transmission of $v_i$, for $1\le i\le n$. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the $n\times n$ matrix whose $(i,j)$-entry is equal to $\frac{{\sigma }_i}{2d_i}+\frac{{\sigma }_j}{2d_j}$ if $v_i$ and $v_j$ are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

This article gives bounds on a substantial number of BID (bond incident degree) indices for connected graphs in terms of their order, size, and maximum degree. The considered BID indices include, among others, the Sombor index (together with its reduced version), atom-bond sum-connectivity index, symmetric division deg index, sum-connectivity index, harmonic index, and Randi´c index. All the graphs that attain the obtained bounds are also characterized. All the established bounds are valid also for molecular graphs. A graph of order n and size m is called an (n,m)-graph. The obtained bounds provide a partial solution to the problem of finding graphs with extremum (considered) BID indices over the class of all connected (n,m)-graphs with a fixed maximum degree under certain constraints.