Narrow Results

Harmonic index is one of the variants of the well-known Randić index. For a graph $G$, the harmonic index of $G$ is defined by the sum of weights $\frac{2}{d(u)+d(v)}$ over all edges $uv$ of $G$, where $d(u)$ stands for the degree of vertex $u$ in $G$. In this paper, an upper bound on the harmonic index of trees with a given order and total domination number is determined and the corresponding extremal trees are characterized. Moreover, our conclusions correct the results presented by Hasni et al., Sharp upper bound for harmonic index of trees with given total domination number, Ars Combin. 159 (2024) 179–186.

The biological, chemical, and structural properties of molecular graphs can be predicted by using topological indices. The lower and upper bounds of fundamental topological indices are well-studied. In this paper, by using comparison theorem for integrals, we obtain new bounds for some topological indices namely, Randić index, Sombor index, harmonic index and forgotten index.

The shunt reactors located on both line terminals and substation bus-bars are commonly used on long extra high voltage (EHV) transmission systems for controlling voltage during load variations. In a small power system that appears in an early stage of a black start of a power system, an overvoltage could be caused by core saturation on the energization of a shunt reactor with residual flux. The most effective method for the limitation of the switching overvoltages is controlled switching since the magnitudes of the produced transients are strongly dependent on the closing instants of the switch. A harmonic index has been introduced that it's minimum value is corresponding to the best-case switching time. In addition, in this paper an artificial neural network (ANN) is used to estimate the optimum switching instants for real time applications. ANN is trained with equivalent circuit parameters of the network, so that developed ANN is applicable to every studied system. To verify the effectiveness of the proposed index and accuracy of the ANN-based approach, two case studies are presented and demonstrated.

Let $v_1,v_2$ be vertices of a graph $G$ with degree of the vertices being $d(v_1)$ and $d(v_2),$ respectively. First, let us define the weight of the edge $v_1{v}_2$ as twice the value of $\frac{1}{d(v_1)+d(v_2)}$ in $G$. Let us define $H(G)$, the harmonic index of the graph $G$, as the sum obtained by adding the weight assigned to every edge of $G$. In this paper, for the class of trees, we shall obtain an upper bound for the harmonic index $H(G)$ in terms of the edge-vertex domination number and the order of $G$. Also, we shall ascertain that the equality is true by characterizing the collection of all extremal trees attaining this bound.

The harmonic index of a graph $G$, denoted by $H(G)$, is defined as the sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [WSEAS Trans. Math.12 (2013) 716–726] proved that for any unicyclic graph $G$ of order $n\ge 3$, $H(G)\le \frac{n}{2}$ with equality if and only if $G=C_n$. Recently, Zhong and Cui [Filomat29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph $G$ of order $n\ge 4$ other than $C_n$, $H(G)\le \frac{n}{2}-\frac{2}{15}$. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph $G$ of order $n\ge 3$ with maximum degree $\mathrm{\Delta }$,

A recently proposed conjecture by Cheng and Wang [A lower bound for the harmonic index of a graph with minimum degree at least three, Filomat30(8) (2016) 2249–2260] about the harmonic index is disproved.

For a simple and connected graph $G$ with $n$ vertices and the minimum degree two, we show that $H(G)\ge 4+\frac{1}{n-1}-\frac{12}{n+1}$ by a technique based on linear programming, where $H(G)$ is the harmonic index of a graph $G$, defined as the sum of the weights $\frac{2}{d_u+d_v}$ of all edges $uv$ of $G$, $d_u$ denotes the degree of a vertex $u$, and characterize the graph with the minimum value.

A graph invariant is any function on a graph that does not depend on a labeling of its vertices. One of the best known graph invariants successfully applied in chemical graph theory over the last decade is the harmonic index. It is defined for a graph $G$ as the sum of the terms $\frac{2}{d_G(u)+d_G(v)}$ over all edges $uv$ of $G$, where $d_G(u)$ and $d_G(v)$ denote the degrees of the vertices $u$ and $v$ in $G$, respectively. The eccentric version of harmonic index has recently been proposed in an analogous way by replacing the vertex degrees with the vertex eccentricities. One of the main topics in chemical graph theory is to study how certain invariants of product graphs are related to the corresponding invariants of their components. Due to this, we investigate here the behavior of the eccentric version of harmonic index under various families of graph products and apply the derived results on some graphs of chemical and general interest.

The harmonic index of graph $G$ is the value $H(G)={\sum }_{uv\in E(G)}\frac{2}{d_u+d_v}$, where $d_x$ refers to the degree of $x$. Zhong [The harmonic index for graphs, Appl. Math. Lett.25 (2012) 561–566] proved that $H(T)\ge \frac{2(n-1)}{n}$ for any tree $T$ of order $n$. As a results of Ali, Raza and Bhatti [Some vertex-degree-based topological indices of cacti, Ars Combin.144 (2019) 195–206], it can be shown that $H(T)\ge \frac{2(n-1)}{\ell +1}$ for any tree $T$ of order $n$ with $\ell$ leaves, where $3\le \ell \le n-2$. In this paper, we generalize this lower bound for all cactus graphs. We present a lower bound for the harmonic index of cactus graphs $G$ in terms of the order, the number of leaves and the number of cycles, and characterize all cactus graphs achieving equality for the given bound.