the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

Recently, Gutman defined a new graph invariant which is named the Sombor index SO(G) of G and computed via expression = ?u~v? qdeg(u)2 + deg(v)2, where deg(u) represents degree vertex u in summing performed across all unordered pairs adjacent vertices v. Damnjanovic et al. have implemented an earlier result obtained by Wang order to show that, among trees TD that specified sequence D, greedy tr...

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Zhang et al (Discrete Appl. Math., 92(1999), 71-84) characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by Gutman (J. Math. Chem., 1(1987), 123-143). In this letter, for a given positive integer d we characterize t...

The graphs having the maximum value of certain bond incident degree indices (including second Zagreb index, general sum-connectivity and zeroth-order Randi? index) in class all connected with fixed order number pendent vertices are characterized this paper. problem finding minimum values index from aforementioned is also addressed. One obtained results about first has already been proved papers...

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological and denoted SO: SO=SO(G)=?vivj?E(G)dG(vi)2+dG(vj)2, where dG(vi) degree vertex vi G. Here, we present novel lower upper bounds on graphs using some parameters. Moreover, obtain several relations with first second Zagreb indices graphs. Finally, give conclusion...

The conditions required for the calcification in vitro of slices of rachitic rat cartilage have been extensively investigated, and attempts have been made to analyse the process in biochemical terms. The study of the role of enzymes in the calcification was initiated by Robison (1923), who discovered high concentrations of alkaline phosphatase in the pre-osseous cartilage of young rats. Robison...

Plasma acid phosphatase estimation is of established value in the diagnosis of carcinoma of the prostate with bone secondaries, and in the assessment of stilboestrol treatment (Gutman and Gutman, 1938; Herbert, 1946; Wray, 1956). Modifications of the method of Gutman and Gutman (1938), using disodium phenyl phosphate substrate, and estimating the phenol liberated with Folin and Ciocalteu reagen...

In this letter, we present an additively manufactured half-Gutman lens antenna operating at 30 GHz. The hemispherical allows for a compact beamformer while maintaining the wide scanning capabilities of Gutman lenses. This solution further enables to better integrate feed system when compared more conventional half-Luneburg design as focal arc is moved inside lens. graded-index implemented with ...