Maximum and Minimum Energy Trees with Two and Three Branched Vertices (2015)
The energy of a graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper we find trees with minimal and maximal energy over the sets Ω (n, 2) and Ω (n, 3) of all trees with n vertices and exactly two and three branched vertices, respectively (see Figures 1 and 2). We show that S (2, 2; n − 8; 2, 2) has maximal energy and S(1, . . . , 1 | {z } n−4 ; 2; 1, 1) has minimal energy over Ω (n, 2). We also find the extremal values of the energy over the set Ωt (n, 2) of all trees in Ω (n, 2) such that the distance between the two branched vertices is exactly t. Finally, we show that among all trees in Ω (n, 3) the tree S(1, 1; 1; 1; 1; 1, . . ., 1| {z } n−6) has minimal energy while the maximal energy is a attained in a tree of the form S(2, 2; p; 2; q; 2, 2)