Article

# On bipartite graphs with minimal energy.

Discrete Applied Mathematics (Impact Factor: 0.72). 01/2009; 157:869-873. 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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ABSTRACT: 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. I. Gutman et al. [MATCH Commun. Math. Comput. Chem. 59, 315–320 (2008; Zbl 1164.05407)] conjectured that the fourth maximal energy tree should be P n (2,6,n-9), which is the tree consisting of three internal disjoint pendent paths starting from the unique vertex of degree 3, the length of the paths are 2,6 and n-9, respectively. Li and Li, Shan and Shao showed that the fourth maximal tree must be one of the two trees, P n (2,6,n-9) and T n (2,2|2,2), and these two trees are incomparable in the so-called quasi-order, where T n (2,2|2,2) denotes the tree of order n obtained by attaching two pendent paths of length 2 to each and vertex of the path P n-8 , respectively. By utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of P n (2,6,n-9) is greater than that of T n (2,2|2,2), and therefore completely confirm this conjecture.
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