Gutman recently introduced a new vertex-degree-based TOPOLOGICAL INDEX called the Sombor INDEX. In this paper, we present some new results relating the Sombor INDEX and some well-studied TOPOLOGICAL indices: Zagreb indices, forgotten INDEX, harmonic INDEX, (general) sum-connectivity INDEX and symmetric division deg INDEX.

Let G be a simple graph with vertex set V (G) and edge set E(G). The multiplicative forgotten TOPOLOGICAL INDEX of G are defined as:  F (G) = Y v2V (G) dG(v)3; where dG(v) is the degree of the vertex v of G. In this paper, we present upper bounds for the multiplicative forgotten TOPOLOGICAL INDEX of several graph operations such as sum, Cartesian product, corona product, composition, strong product, disjunction and symmetric difference in terms of the F􀀀 INDEX and the first Zagreb INDEX of their components. Also, we give explicit formulas for this new graph invariant under two graph operations such as union and Tensor product. Moreover, we obtain the expressions for this new graph invariant of subdivision graphs and vertex – semitotal graphs. Finally, we compare the discriminating ability of indices.

‎In this study‎, ‎we find some bounds for the Sombor INDEX of a graph‎ ‎$G$‎ ‎by some TOPOLOGICAL and statistical indices such as {\it arithmetic INDEX}‎, ‎{\it geometric INDEX}‎, ‎‎ ‎{\it arithmetic-geometric (AG) INDEX}‎, ‎{\it geometric-arithmetic (GA) INDEX}‎, ‎{\it symmetric division degree INDEX (SDD(G))}‎, ‎and some central and dispersion indices‎. ‎The bounds can state estimated values and error intervals for the Somber INDEX and show limits of accuracy‎. ‎Error intervals are expressed as inequalities‎.

Let G be a graph with vertex set V (G) and edge set E(G), and let du denote the degree of vertex u 2 V (G). The Randi c INDEX of G is de ned as R(G) = P 1= p dudv: In this paper, we investigate the relationships between Randi c uv2E(G) INDEX and several TOPOLOGICAL indices.

In [13] Gutman introduced a novel graph invariant called Sombor INDEX SO, defined via $\sqrt{\deg(u)^{2}+\deg(v)^{2}}.$ In this paper we provide relations between Sombor INDEX and some degree-based TOPOLOGICAL indices: Zagreb indices, Forgotten INDEX and Randi\' {c} INDEX. Similar relations are established in the class of triangle-free graphs.

Assume denotes a connected and simple graph with edge set  E(G) as well as vertex set ‎V(G). ‎In chemical graph theory‎, ‎the atom-bond connectivity INDEX, as well as the Randić INDEX of graph are two well-defined TOPOLOGICAL indices‎. ‎In addition‎, ‎Ali and Du [On the difference between  ABC and Randić indices of binary and chemical trees‎, ‎Int‎. ‎J‎. ‎Quantum Chem‎. ‎(2017) e25446] recently unveiled the distinction between Randić and ABC indices‎. ‎In this report‎, ‎we study the link between the difference of Randić and ABC indices with certain well-studied TOPOLOGICAL indices‎.

In this paper we give some characterizations of TOPOLOGICAL extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative TOPOLOGICAL left invariant mean then T is TOPOLOGICAL left lumpy if and only if there is a multiplicative TOPOLOGICAL left invariant mean M on M(S)* such that M(XT)=1, where XT is the characteristic functional of T. Consequently if T is a TOPOLOGICAL left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely TOPOLOGICAL left amenable if and only if S is.

In this paper, a novel TOPOLOGICAL INDEX, named M− INDEX, is introduced based on expanded form of the Wiener matrix. For constructing this INDEX the atomic characteristics and the interaction of the vertices in a molecule are taken into account. The usefulness of the M− INDEX is demonstrated by several QSPR/QSAR models for different physico− chemical properties and biological activities of a large number of diversified compounds. Moreover, the applicability of the proposed INDEX has been checked among isomeric compounds. In each case the stability of the obtained model is confirmed by the cross-validation test. The results of present study indicate that the M− INDEX provides a promising route for developing highly correlated QSPR/QSAR models. On the other hand, the M− INDEX is easy to generate and the developed QSPR/QSAR models based on this INDEX are linearly correlated. This is an interesting feature of the M− INDEX when compared with quantum chemical descriptors which require vast computational cost and exhibit limitations for large sized molecules.

In this paper, we study the separtion axioms T 0, T 1, T 2 and T 5/2 on TOPOLOGICAL and semiTOPOLOGICAL residuated lattices and we show that they are equivalent on TOPOLOGICAL residuated lattices. Then we prove that for every infinite cardinal number a, there exists at least one nontrivial Hausdor TOPOLOGICAL residuated lattice of cardinality a. In the follows, we obtain some conditions on (semi) TOPOLOGICAL residuated lattices under which this spaces will convert into regular and normal spaces.Finally by using of regularity and normality, we convert (semi) TOPOLOGICAL residuated lattices into metrizable TOPOLOGICAL residuated lattices.