Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solution to a conjecture on the maximal energy of bipartite bicyclic
  graphs   The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Let $C_n$
denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joining
two cycles $C_6$ by a path $P_{n-12}$ with its two leaves. Let $\mathscr{B}_n$
denote the class of all bipartite bicyclic graphs but not the graph $R_{a,b}$,
which is obtained from joining two cycles $C_a$ and $C_b$ ($a, b\geq 10$ and $a
\equiv b\equiv 2\, (\,\textmd{mod}\, 4)$) by an edge. In [I. Gutman, D.
Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computer
experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and
Vidovi\'{c} conjectured that the bicyclic graph with maximal energy is
$P^{6,6}_n$, for $n=14$ and $n\geq 16$. In [X. Li, J. Zhang, On bicyclic graphs
with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and
Zhang showed that the conjecture is true for graphs in the class
$\mathscr{B}_n$. However, they could not determine which of the two graphs
$R_{a,b}$ and $P^{6,6}_n$ has the maximal value of energy. In [B. Furtula, S.
Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy,
{\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations
up to $a+b=50$ were reported, supporting the conjecture. So, it is still
necessary to have a mathematical proof to this conjecture. This paper is to
show that the energy of $P^{6,6}_n$ is larger than that of $R_{a,b}$, which
proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic
graphs, the conjecture is still open.

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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn