Volume 11, Issue 1 (4-2016)                   IJMSI 2016, 11(1): 115-122 | Back to browse issues page

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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 dfrac{1}{2}+dfrac{2}{3(n-2)}$, where $n$ is the order and $D(G)$ is the diameter of the graph $G$.

Type of Study: Research paper | Subject: Special