Minimal Energy on Unicyclic Graphs 
Received:July 10, 2013 Revised:February 24, 2014 
Key Word:
graph energy unicyclic graph matching quasiorder.

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). 

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Abstract: 
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 $C_{9}^{3}(2,2,2)\circ S_{n8}$ 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 $r=3,4$. 
Citation: 
DOI:10.3770/j.issn:20952651.2014.04.004 
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