arXiv subjects

Aleksandar Ilić

Publications and source records attributed to Aleksandar Ilić.

8 recordsLinked to original sources

arXiv2026

Further results on the lower bound on reduced Zagreb index of trees

For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_λ(G) = \sum_{uv \in E} (deg(u) + λ) (deg(v) + λ),$$ where $λ$ is an arbitrary real number and $deg (v)$ is the degree of the vertex $v$. In this paper, we extend and correct the equality results from [N. Dehgardia, S. Klav\v zar, {\it Improved lower bounds on the general reduced second Zagreb index of trees}, preprint (2023)] regarding the minimal value of $GRM_λ$ for $λ\geq -1$ among trees with $n$ vertices and a maximal degree $Δ$. Furthermore, we complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with $Δ= 3$ and $Δ= 4$ in $λ= -2$, and characterize the extremal trees.

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arXiv2023

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 $D \subseteq \{d : d \mid n,\ 1\leq d 1,\ 1\leq i\leq k \}|$, 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.

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arXiv2021

On the computational complexity of the Steiner $k$-eccentricity

The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V (G)$ which contain $v$. A linear time algorithm for calculating the Steiner $k$-eccentricity of a vertex on block graphs is presented. For general graphs, an $O(n^{ν(G)+1}(n(G) + m(G) + k))$ algorithm is designed, where $ν(G)$ is the cyclomatic number of $G$. A linear algorithm for computing the Steiner $3$-eccentricities of all vertices of a tree is also presented which improves the quadratic algorithm from [Discrete Appl.\ Math.\ 304 (2021) 181--195].

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arXiv2020

Comparing Wiener complexity with eccentric complexity

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. The Wiener complexity of $G$ is the number of different transmissions of its vertices. Similarly, the eccentric complexity of $G$ is defined as the number of different eccentricities of its vertices. In this paper these two complexities are compared. The complexities are first studied on Cartesian product graphs. Transmission indivisible graphs and arithmetic transmission graphs are introduced to demonstrate sharpness of upper and lower bounds on the Wiener complexity, respectively. It is shown that for almost all graphs the Wiener complexity is not smaller than the eccentric complexity. This property is proved for trees, the equality holding precisely for center-regular trees. Several families of graphs in which the complexities are equal are constructed. Using the Cartesian product, it is proved that the eccentric complexity can be arbitrarily larger than the Wiener complexity. Additional infinite families of graphs with this property are constructed by amalgamating universally diametrical graphs with center-regular trees.

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arXiv2011

On comparing Zagreb indices

Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j) \in E} d_i d_j$, where $d_i$ denote the degree of vertex $i$. Recently proposed conjecture $M_1 / n \leqslant M_2 / m$ has been proven to hold for trees, unicyclic graphs and chemical graphs, while counterexamples were found for both connected and disconnected graphs. Our goal is twofold, both in favor of a conjecture and against it. Firstly, we show that the expressions $M_1/n$ and $M_2/m$ have the same lower and upper bounds, which attain equality for and only for regular graphs. We also establish sharp lower bound for variable first and second Zagreb indices. Secondly, we show that for any fixed number $k\geqslant 2$, there exists a connected graph with $k$ cycles for which $M_1/n>M_2/m$ holds, effectively showing that the conjecture cannot hold unless there exists some kind of limitation on the number of cycles or the maximum vertex degree in a graph. In particular, we show that the conjecture holds for subdivision graphs.

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arXiv2011

On the clique number of integral circulant graphs

The concept of gcd-graphs is introduced by Klotz and Sander, which arises as a generalization of unitary Cayley graphs. The gcd-graph $X_n (d_1,...,d_k)$ has vertices $0,1,...,n-1$, and two vertices $x$ and $y$ are adjacent iff $\gcd(x-y,n)\in D = \{d_1,d_2,...,d_k\}$. These graphs are exactly the same as circulant graphs with integral eigenvalues characterized by So. In this paper we deal with the clique number of integral circulant graphs and investigate the conjecture proposed in \cite{klotz07} that clique number divides the number of vertices in the graph $X_n (D)$. We completely solve the problem of finding clique number for integral circulant graphs with exactly one and two divisors. For $k \geqslant 3$, we construct a family of counterexamples and disprove the conjecture in this case.

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arXiv2011

All pentagonal face multi tori

Design of monomeric unit, by using sequences of map operations, and the growing process in the building of a double-shell multi torus with all-pentagonal faces, is presented. It is shown that the monomer and some small intermediates, as hydrogenated species, have a moderate stability, between adamantane and C60 fullerene, as calculated at the PM3 level of theory. The topology of these high genera structures is described in terms of Omega polynomial as a function of the net parameters. Closed formulas for this polynomial and examples are presented.

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arXiv2011

Eccentric connectivity index

The eccentric connectivity index $ξ^c$ 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 $ξ^c (G) = \sum_{v \in V (G)} deg (v) \cdot ε(v)$\,, where $deg (v)$ and $ε(v)$ 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.

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