Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete solution to a conjecture on the maximal energy of unicyclic
  graphs   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 $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex
of $C_\ell$ with a leaf of $P_{n-\ell}$\,. In [G. Caporossi, D. Cvetkovi\'c, I.
Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding
graphs with extremal energy, {\it J. Chem. Inf. Comput. Sci.} {\bf 39}(1999)
984--996], Caporossi et al. conjectured that the unicyclic graph with maximal
energy is $C_n$ if $n\leq 7$ and $n=9,10,11,13,15$\,, and $P_n^6$ 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
technique, we completely solve this conjecture. However, it turns out that for
$n=4$ the conjecture is not true, and $P_4^3$ should be the unicyclic graph
with maximal energy.

7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph