the harmonic index h(g) , of a graph g is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in e(g), where deg (u) denotes the degree of a vertex u in v(g). in this paper we define the harmonic polynomial of g. we present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in caterpill...

The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpill...

‎the harmonic index of a connected graph $g$‎, ‎denoted by $h(g)$‎, ‎is‎ ‎defined as $h(g)=sum_{uvin e(g)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in g‎. ‎in this paper‎, ‎expressions for the harary indices of the‎ ‎join‎, ‎corona product‎, ‎cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe 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 [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph $G$ of order $ngeq 4$, $H(G)le frac{n}{2}-frac{1}{15}$ and characterize all extremal bicyclic graphs.In this...

A finite subset Y on the unit sphere Sn−1 ⊆ Rn is called a spherical design of harmonic index t, if the following condition is satisfied: ∑ x∈Y f(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of N = {1, 2, · · · }, a finite subset Y ⊆ Sn−1 is called a spherical design of harmonic index T, if ∑ x∈Y f(x) = 0 is satisfied for all real homo...

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

The harmonic index is one of the most important indices in chemical and mathematical fields. It’s a variant of the Randić index which is the most successful molecular descriptor in structure-property and structureactivity relationships studies. The harmonic index gives somewhat better correlations with physical and chemical properties comparing with the well known Randić index. The harmonic ind...