The Harmonic index $ H(G) $ of a graph $ G $ is defined as the sum of the weights $ dfrac{2}{d(u)+d(v)} $ of all edges $ uv $ of $G$, where $d(u)$ denotes the degree of the vertex $u$ in $G$. In this work, we prove the conjecture $dfrac{H(G)}{D(G)} geq dfrac{1}{2}+dfrac{1}{3(n-1)}  $ given by Jianxi Liu in 2013 when G is a unicyclic graph and give a better bound $ dfrac{H(G)}{D(G)}geq dfra...

We study the stability of harmonic morphisms as a subclass of harmonic maps. As a general result we show that any harmonic morphism to a manifold of dimension at least three is stable with respect to some Riemannian metric on the target. Furthermore we link the index and nullity of the composition of harmonic morphisms with the index and nullity of the composed maps.

The harmonic index of a graph G is defined as the sum of weights 2 deg(v)+deg(u) of all edges uv of E(G), where deg(v) denotes the degree of a vertex v in V (G). In this note we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561– 566] and establish some upper and lower bounds on the harmonic index of G.

The harmonic index of a graph G is defined as the sum of weights 2 d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, we determine the graph with minimum harmonic index among all unicyclic graphs with a perfect matching. Moreover, the graph with minimum harmonic index among all unicyclic graphs with a given matching num...