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This paper is concerned with the general atom-bond sum-connectivity index A B S γ , which is a generalization of the recently proposed atom-bond sum-connectivity index, where γ is any real number. For a connected graph G with more than two vertices, the number A B S γ (G) is defined as the sum of (1 − 2 (d x + d y) − 1) γ over all edges x y of the graph G, where d x and d y represent the degrees of the vertices x and y of G, respectively. For − 10 ≤ γ ≤ 10 , the significance of A B S γ is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index A B S γ for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under A B S γ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved. [ABSTRACT FROM AUTHOR]

Let G be a graph containing no isolated vertices. For the graph G, its modified first Zagreb index is defined as the sum of reciprocals of squares of vertex degrees of G. This article provides some new bounds on the modified first Zagreb index of G in terms of some other well-known graph invariants of G. From the obtained bounds, several known results follow directly. [ABSTRACT FROM AUTHOR]

Let G = (V;E), V = fv1; v2;:::; vng, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d1 d2 dn > 0, di = d(vi). Let A = (aij)nn and D = diag(d1; d2;:::; dn) be the adjacency and the diagonal degree matrix of G, respectively. Denote by L+(G) = D-1=2(D + A)D-1=2 the normalized signless Laplacian matrix of graph G. The eigenvalues of matrix L+(G), 2 = + 1 + 2 = + n = 0, are normalized signless Laplacian eigenvalues of G. In this paper some bounds for the sum K+(G) = Pn i=1 1 + i are considered. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n , without isolated vertices. Denote by A = (a i j) n × n the adjacency matrix of G. Eigenvalues of the matrix A , λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n , form the spectrum of the graph G. An important spectrum-based invariant is the graph energy, defined as E (G) = ∑ i = 1 n | λ i |. The determinant of the matrix A can be calculated as det A = ∏ i = 1 n λ i . Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem.77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants E (G) and | det A | are derived. Consequently, all the bounds established in the aforementioned paper are improved. [ABSTRACT FROM AUTHOR]

Let G be a non-trivial connected graph. If the vertices v i and v j of G are adjacent, we write i ∼ j , otherwise we write i ≁ j. Denote by d i the degree of the vertex v i of G. The modified second Zagreb index and coindex of G are defined as M 2 ∗ (G) = ∑ i ∼ j 1 d i d j and M 2 ∗ ¯ (G) = ∑ i ≁ j 1 d i d j , respectively. Several lower and upper bounds on the modified second Zagreb index/coindex are derived and all the graphs attaining these bounds are characterized. The obtained bounds can be used to deduce bounds for various other topological indices. [ABSTRACT FROM AUTHOR]

A general inequality for non-negative real numbers is proven. Based on it, upper bounds for (ordinary) graph energy, minimum dominating energy, minimum covering energy, Laplacian-energy-like invariant, Laplacian energy, Randić energy, and incidence energy are obtained. [ABSTRACT FROM AUTHOR]

Let G be an undirected connected graph with n; n ⩾ 2, vertices and m edges with vertex degrees Δ = d1 ⩾ d2 ⩾ … ⩾ dn =δ. Lower and upper bounds of a graph invariants M1 = Σni =1 d2 i , referred to as the first Zagreb index, and M1 = Σ i×j (di +dj), named the first Zagreb coindex, depending on parameters n, m, Δ and δ are obtained. The obtained results represent improvement of the results reported in the literature. [ABSTRACT FROM AUTHOR]

Let G be an undirected simple and connected graph with n vertices .n ≥ 3/ and m edges. Denote by μ1 ≥ μ2 ≥ ... ≥ μn-1 > μn = 0, γ1 ≥ γ2 ≥ ... ≥ γn, and ρ1 ≥ ρ2 ≥ ... ≥ ρn-1 > ρn = 0, respectively, the Laplacian, signless Laplacian, and normalized Laplacian eigenvalues of G. The Laplacian energy, signless Laplacian energy, and normalized Laplacian energy of G are defined as LE = Σn i=1/μi - 2m/n/, SLE = Σni=1/γi - 2m/n/, and NLE =Σni=1/ρi -1/, respectively. Lower bounds for LE, SLE, and NLE are obtained. [ABSTRACT FROM AUTHOR]

This study aimed to investigate the effect of 150 mg/L sodium fluoride (NaF) on redox status parameters and essential metals [copper (Cu), iron (Fe), and zinc (Zn)] in the blood, liver, kidney, brain, and spleen of Wistar rats and to determine the protective potential of selenium (Se) against fluoride (F-) toxicity. Male Wistar rats were randomly distributed in groups of five (n=5) receiving tap water (control) or water with NaF 150 mg/L, NaF 150 mg/L + Se 1.5 mg/L, and Se 1.5 mg/L solutions ad libitum for 28 days. Fluorides caused an imbalance in the redox and biometal (Cu, Fe, and Zn) status, leading to high superoxide anion (O2.-) and malondialdehyde (MDA) levels in the blood and brain and a drop in superoxide dismutase (SOD1) activity in the liver and its increase in the brain and kidneys. Se given with NaF improved MDA, SOD1, and O2.- in the blood, brain, and kidneys, while alone it decreased SH group levels in the liver and kidney. Biometals both reduced and increased F- toxicity. Further research is needed before Se should be considered as a promising strategy for mitigating F- toxicity. [ABSTRACT FROM AUTHOR]

Vascular complications in patients with diabetes mellitus (DM) are common. Since impaired oxygen balance in plasma plays an important role in the pathogenesis of chronic DM-associated complications, the administration of hyperbaric oxygen therapy (HBOT) has been recommended to influence development of vascular complications. Hyperbaric oxygen therapy involves inhalation of 100% oxygen under elevated pressure from 1.6 to 2.8 absolute atmospheres in hyperbaric chambers. Hyperbaric oxygen therapy increases plasma oxygen solubility, contributing to better oxygen diffusion to distant tissues and preservation of the viability of tissues reversibly damaged by atherosclerosis-induced ischemia, along with microcirculation restoration. Hyperbaric oxygen therapy exerts antiatherogenic, antioxidant, and cardioprotective effects by altering the level and composition of plasma fatty acids and also by promoting signal transduction through membranes, which are impaired by hyperglycemia and hypoxia. In addition, HBOT affects molecules involved in the regulation of nitric oxide synthesis and in that way exerts anti-inflammatory and angiogenic effects in patients with DM. In this review, we explore the recent literature related to the effects of HBOT on DM-related vascular complications. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n and size m, the Laplacian energy of G is defined as LE(G) = where µ1,µ2,..., µn-1; µn are the Laplacian eigenvalues of G satisfying µ1 ≥ µ2 ≥ ... ≥ µn-1 > µn = 0. In this note, some new lower bounds on the graph invariant LE(G) are derived. The obtained results are compared with some already known lower bounds of LE(G). [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph of order n and size m , with vertex set V ( G ) = { v 1 , v 2 , … , v n } , without isolated vertices and sequence of vertex degrees Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ > 0 , d i = d G ( v i ) . If the vertices v i and v j are adjacent, we denote it as v i v j ∈ E ( G ) or i ∼ j . With TI we denote a topological index that can be represented as T I = T I ( G ) = ∑ i ∼ j F ( d i , d j ) , where F is an appropriately chosen function with the property F ( x , y ) = F ( y , x ) . A general extended adjacency matrix A = ( a i j ) of G is defined as a i j = F ( d i , d j ) if the vertices v i and v j are adjacent, and a i j = 0 otherwise. Denote by f i , i = 1 , 2 , … , n the eigenvalues of A . The “energy” of the general extended adjacency matrix is defined as E T I = E T I ( G ) = ∑ i = 1 n | f i | . Lower and upper bounds on E T I are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases. [ABSTRACT FROM AUTHOR]

Two weighted inequalities for real non-negative sequences are proven. The first one represents a generalization of the Szőkefalvi Nagy inequality for the variance, and the second a generalization of the discrete Chebyshev inequality for two real sequences. Then, the obtained inequalities are used to determine lower bounds for some degree-based topological indices of graphs. [ABSTRACT FROM AUTHOR]

The McClelland formula, based on the upper bound $$\sqrt{2mn}$$ , is capable of reproducing over 99.5% of the total $$\pi $$ -electron energy ( $$E_\pi $$ ) of a conjugated hydrocarbon, whose molecules possess n carbon atoms and m carbon-carbon bonds. Its weak point is that it predicts equal $$E_\pi $$ -values for all isomers. We now show how this failure can be overcome, offering a general strategy for extending McClelland's formula. By means of one of these extensions, $$E_\pi $$ is related with the energy of the highest occupied molecular orbital, and the error of the new formula is diminished by more than $$50\%$$ relative to the standard McClelland approximation. [ABSTRACT FROM AUTHOR]

The cabinets for physical therapy are provided with various devices which shorten the time needed for patients' recovery and healing. A typical physical medicine cabinet has tens of stand-alone devices for different purposes. These are the devices which perform biostimulation using diadynamic or interferential electrical currents, ultrasound waves, vacuum impulses, electromagnetic fields, etc. The aim of this paper is to present hardware and software components, and developed network protocol for integrated system of devices for physical medicine, based on the existing products of Elektromedicina company, with key feature that allows the devices to be centrally managed and monitored. This paper addresses the system's architecture, user interfaces both for devices and centralized server console, network protocol for communication between devices and centralized management station, the scheduler, and the therapy procedure within the system. It will be shown that the efficiency of the integrated system, measured in the time-per-patient and patients-per-day, compared with the efficiency of regular physical therapy cabinet is increased by more than 20%. [ABSTRACT FROM AUTHOR]

A relation is obtained between the total π -electron energy E π and the HOMO energy E HOMO , valid within the HMO approximation. This seems to be the very first relation between E π and E HOMO ever established. It enables a much more accurate assessment of E π of alternant conjugated hydrocarbons than that based on McClelland's formula and on other ( n , m )-type approximate expressions. [ABSTRACT FROM AUTHOR]