On Some Topological Indices Defined via the Modified Sombor Matrix.

Let G be a simple graph with the vertex set V={v1,…,vn} and denote by dvi the degree of the vertex vi. The modified Sombor index of G is the addition of the numbers (dvi2+dvj2)-1/2 over all of the edges vivj of G. The modified Sombor matrix AMS(G) of G is the n by n matrix such that its (i,j)-entry is equal to (dvi2+dvj2)-1/2 when vi and vj are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of AMS(G). The sum of the absolute eigenvalues of AMS(G) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is 2; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.

Hosoya Polynomials of Power Graphs of Certain Finite Groups.

Assume that G is a finite group. The power graph P(G) of G is a graph in which G is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order pq and pqr, where p,q and r(p≥q≥r) are prime numbers.

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups.

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric-arithmetic index, and the Randic index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ΓG of a finite group G is a graph where non-central elements of G are its vertex set, while two different elements are edge connected when they do not commute in G. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randic index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric-arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of SL(2,C). We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

Power Graphs of Finite Groups Determined by Hosoya Properties.

Suppose G is a finite group. The power graph represented by P(G) of G is a graph, whose node set is G, and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph Γ is known as the Z-index or Hosoya index. The Z-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.

Sombor spectra of chain graphs.

We study the Sombor index and the Sombor spectral properties of chain graphs. In particular, an explicit formula for the Sombor index is given, the Sombor eigenvalues are discussed, bounds on the largest and the smallest Sombor eigenvalues are presented, chain graphs with the simple Sombor eigenvalue are characterized, formulae for the Frobenius norm and the determinant of the Sombor quotient matrix of chain graphs are given, the Sombor spread bound and the Sombor energy bounds are presented along with the characterization of graphs attaining them.

Synthesis and evaluation of novel 2-substituted-quinazolin-4(3H)-ones as potent analgesic and anti-inflammatory agents.

A novel series of 2-substituted-quinazolin-4(3H)-ones were synthesized by reacting 3,5-disubstituted-anthranilic acid with acetic anhydride/benzoyl chloride, which were further reacted with different primary amines to obtain 2,6,8-substituted-quinazolin-4(3H)-ones 6a-f, 7, 8. All the synthesized compounds were characterized and screened for analgesic and anti-inflammatory activities. Compounds 6,8-dibromo-2-phenyl-3-(4'-carboxyl phenyl)quinazolin-4(3H)-one 7 and 6,8-dibromo-2-phenyl-3-(2'-phenylethanoic acid)quinazolin-4(3H)-one 8 displayed good analgesic and anti-inflammatory activity in comparison to the reference standards acetyl salicylic acid and indomethacin, respectively.

Synthesis and anti-microbial screening of some Schiff bases of 3-amino-6,8-dibromo-2-phenylquinazolin-4(3H)-ones.

In the present study, a series of novel Schiff bases were synthesized by condensation of 3-amino-6,8-dibromo-2-phenylquinazolin-4(3H)-ones with different aromatic aldehydes via cyclized intermediate 6,8-dibromo-2-phenyl benzoxazin-4-one. The chemical structures were confirmed by means of IR, (1)H NMR, (13)C NMR, Mass spectral and Elemental analysis. These compounds were screened for anti-bacterial (Staphylococcus aureus ATCC-9144, Staphylococcus epidermidis ATCC-155, Micrococcus luteus ATCC-4698, Bacillus cereus ATCC-11778, Escherichia coli ATCC-25922, Pseudomonas aeruginosa ATCC-2853, and Klebsiella pneumoniae ATCC-11298) and anti-fungal (Aspergillus niger ATCC-9029 and Aspergillus fumigatus ATCC-46645) activities by paper disc diffusion technique. The minimum inhibitory concentrations (MICs) of the compounds were also determined by agar streak dilution method. Among the synthesized compounds 3-(3,4,5-trimethoxybenzylideneamino)-6,8-dibromo-2-phenylquinazolin-4(3H)-one 10 was found to be the most potent anti-microbial activity with MICs of 18.9, 19.1, 18.8, 21.7, 18.2, 19.3, 16.7, 8.6 and 10.1 microg/ml against above mentioned respective strains. Compounds were found to exhibit more anti-fungal than anti-bacterial activity.