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Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at  x = 1 , we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors. [ABSTRACT FROM AUTHOR]

• The Mostar index is a recently introduced bond-additive distance-based graph invariant that measures the degree of peripherality of particular edges and of the graph as a whole. • This paper aims to gather some known bounds and extremal results concerning the Mostar index. Also, it presents various modifications and generalizations of the aforementioned index and it outlines several possible directions of further research. • Finally, some open problems and conjectures are listed. The Mostar index is a recently introduced bond-additive distance-based graph invariant that measures the degree of peripherality of particular edges and of the graph as a whole. It attracted considerable attention, both in the context of complex networks and in more classical applications of chemical graph theory, where it turned out to be useful as a measure of the total surface area of octane isomers and as a tool for studying topological aspects of fullerene shapes. This paper aims to gather some known bounds and extremal results concerning the Mostar index. Also, it presents various modifications and generalizations of the aforementioned index and it outlines several possible directions of further research. Finally, some open problems and conjectures are listed. [ABSTRACT FROM AUTHOR]