Eccentric connectivity index

The eccentric connectivity index $\xi^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modelled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph $ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, \ldots, n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. Motivated by the result on the upper bound of the diameter of $ICG_n(D)$ given in [N. Saxena, S. Severini, I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic quantum dynamics}, International Journal of Quantum Information 5 (2007), 417--430], according to which $2|D|+1$ represents one such bound, in this paper we prove that the maximal value of the diameter of the integral circulant graph $ICG_n(D)$ of a given order $n$ with its prime factorization $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, is equal to $r(n)$ or $r(n)+1$, where $r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|$, depending on whether $n\not\in 4N+2$ or not, respectively. Furthermore, we show that, for a given order $n$, a divisor set $D$ with $|D|\leq k$ can always be found such that this bound is attained. Finally, we calculate the maximal diameter in the class of integral circulant graphs of a given order $n$ and cardinality of the divisor set $t\leq k$ and characterize all extremal graphs. We actually show that the maximal diameter can have the values $2t$, $2t+1$, $r(n)$ and $r(n)+1$ depending on the values of $t$ and $n$. This way we further improve the upper bound of Saxena, Severini and Shparlinski and we also characterize all graphs whose diameters are equal to $2|D|+1$, thus generalizing a result in that paper.Comment: 29 pages, 1 figur

On the sum of powers of Laplacian eigenvalues of bipartite graphs

summary:For a bipartite graph $G$ and a non-zero real $\alpha$, we give bounds for the sum of the $\alpha$th powers of the Laplacian eigenvalues of $G$ using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced

Note on PI and Szeged indices

In theoretical chemistry molecular structure descriptors are used for modeling physico-chemical, pharmacological, toxicologic, biological and other properties of chemical compounds. In this paper we study distance-based graph invariants and present some improved and corrected sharp inequalities for PI, vertex PI, Szeged and edge Szeged topological indices, involving the number of vertices and edges, the diameter, the number of triangles and the Zagreb indices. In addition, we give a complete characterization of the extremal graphs.Comment: 10 pages, 3 figure

O modernizmu i avangardi u Jugoslaviji

Biljana Andonovska, Tomislav Brlek, Adrijana Vidić (ur.), Modernizam i avangarda u jugoslovenskom kontekstu I–II, Beograd: Institut za književnost i umetnost, 2021/2022, 222 str. I. i 188 str. II

O modernizmu i avangardi u Jugoslaviji

Biljana Andonovska, Tomislav Brlek, Adrijana Vidić (ur.), Modernizam i avangarda u jugoslovenskom kontekstu I–II, Beograd: Institut za književnost i umetnost, 2021/2022, 222 str. I. i 188 str. II

Numerical analyses of water hammer and water-mass oscillations in a hydropower plant for the most extreme operational regimes

Analize hidrauličnih prelaznih radnih režima su neophodne u fazi projektovanja novih i revitalizacije postojećih hidroelektrana. U ovom radu su razmatrani prelazni procesi pri specifičnim ekstremnim radnim režimima za derivacionu hidroelektranu u kojoj su ugrađeni vodostan i sinhroni regulator pritiska na spiralnom kućištu turbine. Posebni osvrti su na analizama prelaznih režima pri oscilacijama vodenih masa i hidrauličkom udaru. Razmatrani su razni eksploatacioni režimi i različiti zakoni rada sihronih regulatora pritiska. Rezultati su dobijeni pomoću originalnog softvera razvijenog za potrebe ovih analiza. Urađena je kalibracija modela, a rezultati su upoređeni sa analizama prelaznih režima iz faze projektovanja postojeće hidroelektrane.Hydraulic transients analyses are necessary during the design stage of both new and refurbished hydropower plants (HPPs). In this paper, transients of specified most extreme operational regimes are investigated for a long derivation system, provided with a surge tank as well as pressure relief valves (PRVs) at the turbines spiral casing. The transients analyses are focused on water-mass oscillations and water hammer. Investigations for various exploitation regimes and different operating laws of the PRV's are adopted. Results are obtained by means of an original software developed for these analyses. The model was duly calibrated, and the results were compared with the results of the transient analyses from the original design phase of the existing HPP