Zagreb indices are among the foremost topological indices in mathematical chemistry. These indices are crucial for investigating the total it-electron energy of alternant hydrocarbons and are utilized to study various aspects of molecular properties, including complexity, chirality, ZE-isomerism, and hetero-systems. In this paper, we focus on two well-known modifications of these indices: the first and second hyper-Zagreb indices. For a finite simple graph Gamma, these indices are expressed as X HM1(Gamma) = 9 is an element of e(Gamma) X (d Gamma (& vartheta;) + d Gamma (omega))2 and HM2(Gamma) = 9 is an element of e(Gamma) (d Gamma(& vartheta;)d Gamma(omega))2, where E(Gamma) denotes the edge set of Gamma and d Gamma(& vartheta;) indicates the degree of the vertex & vartheta; in Gamma. In this paper, we introduce graph transformations on trees and connected graphs that minimize the first and second hyper-Zagreb indices. Accordingly, we determine the minimum values of these indices within the class of all trees with a given number of vertices and a specified maximum vertex degree. Additionally, we characterize the corresponding minimal trees. Our results will be extended to all connected graphs with a given order and maximum vertex degree.
A Roman dominating function (RDF) on a graph G is a function f : V (G) -> {0, 1, 2} such that any vertex v with f (v) = 0 is adjacent to at least one vertex w with f(w) = 2. In addition, if the subgraph of G induced by the set of all vertices for which f not equal 0 has no isolated vertices then f is called a total Roman dominating function (TRDF). If f is an RDF (TRDF) then the least value of Sigma(u is an element of V) f(u) is called Roman domination number (total Roman domination number) of G and is denoted by 7R (G) ( 7t(R) (G)). Let G = G(n, k, 0) be a Kneser graph, where n, k are positive integers. In this paper we present some bounds for 7tR(G(n, k, 0)) for k(2) < n < k(2) + k. In particular we show that 7tR(G(k(2) + k 1, k, 0)) = 2(k + 2) and for n > 2k + 1, 7R(G(n, k, 0)) >= max{7R(G(n 1, k 1, 0)), 7(R) (G(n, k 1, 0))}.
Let N and De be net Laplacian and net distance matrices of a signed tree, respectively. The inverse (resp. group inverse) of De is obtained if it is nonsingular (resp. singular), which extend the inverse formula obtained by Graham and Lovasz for distance matrix of a unsigned tree. The interlacing inequality connecting the eigenvalues of De and N of a signed tree is also obtained.
For centuries, Iranian philosophers and scientists believed that music is related to sciences such as mathematics, the four temperaments of humans, and astronomy; unfortunately, none of them provided explicit rules or mathematical formulations to substantiate their claims. Addressing the gap, this article introduces a 17-note numerical system along with a computational method designed to identify the psychological effects and emotional states associated with musical scales. New concepts, specifically the "innate scale" and "acquired scale," are introduced with their distinctions and applications explored in both musical and numerical frameworks. Additionally, the study applies computational techniques and principles of numerical analysis to investigate the primary modes of the 7 Dastgah system in Classical Persian Music. Although the 17-note system shares interval similarities with the ancient Abjad 18-note system, it differs substantially in numerical valuation, computational methodology, structural efficiency, and systematic approach to application. The findings can be applied to music therapy and theoretical music studies as an innovative, numerically grounded method that offers deeper insights into the psychological attributes of musical scales. The results reveal a significant correlation between the psychological states induced by musical scales and their innate temperament qualities, and by identifying the zodiac constellations associated with each scale, the psychological impacts of melodies can be further discerned. For instance, analysis of the primary mode in Dastgah-e Chahargah indicates that feelings of excitement align with the psychological quality of warm and dry. Furthermore, the computational approach revealed Chahargah astrological with Leo, suggesting that its melodies naturally evoke qualities of courage, exhilaration, and radiant joy.
Let G be a graph with order n. Aouchiche and Hansen first proposed the distance Laplacian matrix of G, defined as L(G) = diag(Tr) - D(G), where D(G) is the distance matrix and diag(Tr) = diag(Tr(v(1)), Tr(v(2)),..., Tr(v(n))) is the diagonal matrix of the vertex transmissions of G, and the largest eigenvalue of L(G) is called the distance Laplacian spectral radius of G, written as rho(L)(G). By using the equitable quotient matrix of L(G), Tutte Theorem and Tutte-Berge Formula of integer k-matching, we establish the lower bound for the distance Laplacian spectral radius of G among all n-vertex graphs with given integer k-matching number and characterized the corresponding extremal graph. This generalizes the results of Wang et al. [Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number, Linear Algebra Appl., 506 (2016) 579- 587.] and Liu et al. [Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of fractional matching number,J. Oper. Res. Soc. China., (2023) 1-8.].
