Article

# On bipartite graphs with minimal energy.

Discrete Applied Mathematics 01/2009; 157:869-873. DOI: 10.1016/j.dam.2008.07.008

Source: DBLP

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**ABSTRACT:**For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal energy tends to be complicated for a given class of graphs. There are many approaches on the maximal energy of trees, unicyclic graphs and bicyclic graphs, respectively. In this paper, we study the maximal energy of tricyclic graphs. Let $P^{6,6,6}_n$ denote the graph with $n\geq 20$ vertices obtained from three copies of $C_6$ and a path $P_{n-18}$ by adding a single edge between each of two copies of $C_6$ to one endpoint of the path and a single edge from the third $C_6$ to the other endpoint of the $P_{n-18}$. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P. Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it EURO J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following conjecture: let $G$ be a tricyclic graphs on $n$ vertices with $n=20$ or $n\geq22$, then $\mathcal{E}(G)\leq \mathcal{E}(P_{n}^{6,6,6})$ with equality if and only if $G\cong P_{n}^{6,6,6}$. We partially solve this conjecture.Match (Mulheim an der Ruhr, Germany) 08/2014; 72(1):183-214. · 1.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this work, a sandwich-type electrochemical aptasensor for simultaneous sensitive detection of platelet-derived growth factor (PDGF) and thrombin is fabricated. Reduced graphene oxide sheets (rGS) are used as matrices to immobilize the redox probes, which are subsequently coated with platinum nanoparticles (PtNPs) to form the PtNPs-redox probes-rGS nanocomposites. With the employment of the as prepared nanocomposites, a signal amplification strategy was described based on bienzyme (glucose oxidase and horseradish peroxidase) modified PtNPs-redox probes-rGS nanocomposites as the tracer labels for secondary aptamers (Apt II) through sandwiched assay. Gold nanoparticles functionalized single-walled carbon nanotubes (AuNPs@SWCNTs) as the biosensor platform enhance the surface area to capture a large amount of primary aptamers (Apt I), thus amplifying the detection response. The experiment results show that the multi-labeled PtNPs-redox probes-rGS nanocomposites display satisfying electrochemical redox activity and highly electrocatalytic activity of PtNPs and bienzyme, which exhibit high sensitivity for detection of proteins. The linear range of PDGF is 0.01-35 nM with a detection limit of 8 pM, while the linear ranges from 0.02 to 45 nM and a detection limit of 11 pM for thrombin are obtained.Biomaterials 11/2011; 33(4):1090-6. · 8.31 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For a given simple graph G, the energy of G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let be the unicyclic graph obtained by connecting a vertex of Cℓ with a leaf of Pn−ℓ. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is Cn if n≤7 and n=9,10,11,13,15, and for all other values of n. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for n=4 the conjecture is not true, and should be the unicyclic graph with maximal energy.European Journal of Combinatorics 08/2011; 32(5):662-673. · 0.66 Impact Factor

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