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On bipartite graphs with minimal energy

Discrete Applied Mathematics (Impact Factor: 0.8). 02/2009; 157(4):869-873. DOI: 10.1016/j.dam.2008.07.008
Source: DBLP

ABSTRACT

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. In a paper [G. Caporossi, D. Cvetkovi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996] Caporossi et al. conjectured that among all connected graphs G with n≥6 vertices and n−1≤m≤2(n−2) edges, the graphs with minimum energy are the star Sn with m−n+1 additional edges all connected to the same vertices for m≤n+⌊(n−7)/2⌋, and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. The conjecture is proved to be true for m=n−1,2(n−2) in the same paper by Caporossi et al. themselves, and for m=n by Hou in [Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163–168]. In this paper, we give a complete solution for the second part of the conjecture on bipartite graphs. Moreover, we determine the graph with the second-minimal energy in all connected bipartite graphs with n vertices and m(n≤m≤2n−5) edges.

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Available from: Xueliang Li, Jan 12, 2014
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    • "This conjecture is true when e = n − 1, 2(n − 2) [1], and when e = n for n ≥ 6 [9]. Li et al. [15] showed that B n,e is the unique bipartite graph of order n with minimal energy for e ≤ 2n − 4. Hou [10] proved that for n ≥ 6, B n,n+1 has the minimal energy among all bicyclic graphs of order n with at most one odd cycle. Let G n,e be the set of connected graphs with n vertices and e edges. "
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    ABSTRACT: The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper, we characterize the tetracyclic graph of order $n$ with minimal energy. By this, the validity of a conjecture for the case $e=n+3$ proposed by Caporossi et al. \cite{CCGH} has been confirmed.
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    • "It is quite interesting to study the extremal values of the energy among some given classes of graphs, and characterize the corresponding extremal graphs. In the meantime , a large number of results were obtained on the minimal energies for distinct classes of graphs, such as acyclic conjugated graphs [25] [32], bipartite graphs [30], unicyclic graphs [13] [23], bicyclic graphs [14], tricyclic graphs [26] [27] and tetracyclic graphs [24]. "
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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.
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    • "By these results, finding unicyclic , bicyclic, and tricyclic graphs with maximum matching energy is an elementary task. In [19], Li and Zhang considered the minimal energy graph among connected bipartite (n, m)graphs . With regard to skew energy, Gong et al. [6] determined the extremal graph in (n, m)-graphs. "
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    ABSTRACT: Let G be a simple graph of order n and μ1, M 2,⋯. μn the roots of its matching polynomial. Recently, Gutman and Wagner defined the matching energy as the sum Σ ni=1 |μi|. In this paper, we first show that Turán graph Tr, n is the r-partite graph of order n with maximum matching energy. Then we characterize the connected graphs (and bipartite graph) of order n having minimum matching energy with m (n + 2 ≤ m ≤ 2n - 4) (n ≤ m ≤ In - 5) edges.
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