Many graphs are constructed from simpler ones by the use of operations on graphs, and as a consequence, the properties of the resulting constructions are strongly related to the properties of their constituents. This paper is concerned with computing some distance-based graph invariants for three constructions on graphs namely double graph, extended double cover, and strong double graph.

Let G and H be two graphs. The strong product G  H of the graphs G and H is the graph with vertex set V (G)  V (H), and u = (u1; v1) is adjacent with v = (u2; v2) whenever (v1 = v2 and u1 is adjacent with u2) or (u1 = u2 and v1 is adjacent with v2) or (u1 is adjacent with u2 and v1 is adjacent with v2). In this paper, some applications of this product are presented. Finally, we pose one open problem related to this topic.

To date, the feasibility of constructing a complete graph invariant in polynomial time remains uncertain. Therefore, developing fast algorithms for checking non-isomorphism, including heuristic ones, is crucial. Successful implementation of these heuristics involves modifying existing graph invariants and creating new ones, both of which are still pertinent. Many existing invariants enable the distinction of a large number of graphs in real time.This paper introduces an invariant specifically for tournaments, a type of directed graph. Tournaments are interesting because the number of different tournaments, given a fixed order of vertices, matches the number of undirected graphs with the same fixed order. The proposed invariant considers all possible tournaments formed by subsets of vertices from the given digraph with the same set of arcs. For each subset tournament, standard places are calculated and summed to determine the final vertex points, which constitute the new invariant.Our calculations reveal that the new invariant differs from the most natural tournament invariant, which assigns points to each participant. Initial computational experiments show that the smallest pair correlation between sequences representing these two invariants is observed at dimension 15.

The transmission of a vertex ${\varsigma}$ in a connected graph $\mathcal{J}$ is the sum of distances between ${\varsigma}$ and all other vertices of $\mathcal{J}$. A graph $\mathcal{J}$ is called transmission regular if all vertices have the same transmission. In this paper, we propose a new graph invariant for measuring the transmission irregularity extent of transmission irregular graphs. This invariant which we call the total transmission irregularity number (TTI number for short) is defined as the sum of the absolute values of the difference of the vertex transmissions over all unordered vertex pairs of a graph. We investigate some lower and upper bounds on the TTI number which reveal its connection to a number of already established indices. In addition, we compute the TTI number for various families of composite graphs and for some chemical graphs and nanostructures derived from them.

‎For any graph $G$‎, ‎the modified Sombor index is defined as the reciprocal of the well-known Sombor Index‎. ‎The girth of $G$‎, ‎by short $g(G)$‎, ‎is the length of the smallest cycle in $G$‎. ‎A graph with exactly one cycle is a unicyclic graph‎. ‎If it is further‎, ‎connected‎, ‎it is a connected unicyclic graph‎. ‎In this article‎, ‎we achieved the modified Sombor index for a collection of graphs‎, ‎connected unicyclic graphs and also obtained the maximum modified Sombor index for the class of unicyclic graphs based on the restrictions by a fixed girth‎.