Minimal Energy on Unicyclic Graphs
Received:July 10, 2013  Revised:February 24, 2014
Key Word: graph energy   unicyclic graph   matching   quasi-order.  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11326216) and the Docter Foundation of Shandong University of Technology (Grant No.413010).
Author NameAffiliation
Shengjin JI School of Science, Shandong University of Technology, Shandong 255049, P. R. China 
Yongke QU Department of Mathematics, Luoyang Normal University, Henan 471022, P. R. China 
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      For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $\mathscr{U}_{n}$ denote the set of all connected unicyclic graphs with order $n$, and $\mathscr{U}^{r}_{n}=\{G\in\mathscr{U}_{n}|\,d(x)=r$ for any vertex $x\in V(C_{\ell})\}$, where $r\geq 2$ and $C_{\ell}$ is the unique cycle in $G$. Every unicyclic graph in $\mathscr{U}^{r}_{n}$ is said to be a cycle-$r$-regular graph. In this paper, we completely characterize that $C_{9}^{3}(2,2,2)\circ S_{n-8}$ is the unique graph having minimal energy in $\mathscr{U}_{n}^{4}$. Moreover, the graph with minimal energy is uniquely determined in $\mathscr{U}_{n}^{r}$ for $r=3,4$.
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