Abstract
For a graph \(G=(V_G,E_G)\), a subset \(S\subseteq V_G\) is called a maximum dissociation set if the induced subgraph \(G[S]\) does not contain \(P_3\) as its subgraph, and the subset has maximum cardinality. The dissociation number of \(G\) is the number of vertices in a maximum dissociation set of \(G\). This paper mainly studies the problem of determining the maximum or minimum values of the Harary indices among all trees, bipartite graphs and general connected graphs with fixed order and dissociation number. To be specific, we determine the sharp upper bound of the Harary index among all connected graphs (resp. bipartite graphs, trees) with given order and dissociation number. The extremal graphs meeting these upper bounds are fully characterized. Furthermore, the graphs having the minimum Harary indices with fixed order \(n\) and dissociation number \(\varphi\in\{2,\left\lceil\frac{2}{3}n\right\rceil,n-2,n-1\}\) are also showed.
