In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected ork -connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.

Let G be a connected graph, and let D[G] denote the double graph of G. In this paper, we first derive closed-form formulas for some distance based topological indices for D[G] in terms of G. Finally, these formulas are applied for several special kinds of graphs, such as, the complete graph, the path and the cycle.

The n-star graph Sn is defined on the set of all n sequenses (u1, u2, . . ., un), ui ∈ {1, 2, . . ., n}, ui \ne uj and i \ne j, where edges are of the form (u1, u2, . . ., un) ∼ (ui, u2, . . ., un), for some i \ne 1. In this paper we will show that Sn is a vertex and edge transitive graph and discuss some topological properties of Sn.

Let a topological space. An intersection graph on a topological space, which denoted by , is an undirected graph which whose vertices are open subsets of and two vertices are adjacent if the intersection of them are nonempty. In this paper, the relation between topological properties of and graph properties of are investigated. Also some classifications and representations for the graph are introduced and at last some results about dominating sets on are explained.

In this work, Mhr- polynomial of oxybenzone, menthyl anthranilate, benzophenone-4, and dimethylamino hydroxybenzoyl hexa benzoate were established. The degree-based topological indices HDR version of Modified Zagreb topological index (HDRM*), HDR version of Modified forgotten topological index (HDRF*), and HDR version of hyper Zagreb index (HDRHM*) were obtained. Accordingly, by using the derivative of Mhr- polynomial of sunscreens, the HDRM*, HDRF*, and HDRHM* topological indices of sunscreens were found.

Let $G$ be a graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).\[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, topological index, toroidal fullerenes.

This paper studies the concept of fuzzy generalized topologies, which are generalizations of smooth topologies and Chang's fuzzy topologies. A basis of fuzzy generalized topological space will be defined as functions from the family of all fuzzy subsets of a non-empty set $ X $ to $ [0, 1] $ and  some basic properties of their structure will be obtained. Some characterizations of  the basis of fuzzy generalized topology, fuzzy generalized cotopology and the product of fuzzy generalized topological spaces will also be shown.

topological indices are extremely useful for analyzing various physical and chemical properties associated with a chemical compound. A topological index describes molecular structures by converting them into certain real numbers. topological indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity of molecule correlated with their chemical structure. The chemical shape of benzene molecule is very common in nano-science, chemistry, and physics. The circumcoronene collection of benzenoid  generates from the benzene molecules. Jahangir graph  is a generalized wheel graph that consists of  circular vertices and a center vertex connected to every 𝑚𝑡 vertex on the circle. In this article, we will compute the topological indices of the middle graph of the circumcoronene series of benzenoids  and Jahangir graph . In addition, comparison of the middle graph of the circumcoronene series of benzenoid  and Jahangir graph  are presented numerically and graphically.