The energy of trees with diameter five under given conditions (Q2089173)

Defined by \textit{I. Gutman} [The energy of a graph. Ber. Math.-Stat. Sekt. Forschungszent. Graz 103, 22 S. (1978; Zbl 0402.05040)] half a century ago, the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Since then, many papers have characterised the structure of trees with various extremal or near-extremal energies. The present paper refines and extends studies of the relative energies of trees with diameter 5 in the papers [\textit{X. Qiao} and \textit{B. Huo}, ``Extremal energy of a class of tree with diameter five'', J. Qinghai Normal Univ. (Nat. Sci.), 2, 29--32 (2018); \textit{Y. Jia} and \textit{B. Huo}, ``The ordering of energies in a class of trees with diameter five'', ibid. 2 (2019)]. In particular, the present paper restricts these studies to a narrower class of trees of diameter~5 that contains the diameter 5 trees with maximal energy. The present authors provide details of the quasi-order on that class defined by the energy and show that the graphs in the class with larger energy have the properties that if two non-center vertices are adjacent to the same center, then the numbers of their pendent neighbors differ by at most one; whereas if two non-center vertices are adjacent to different centers, then the numbers of their pendent neighbors differ by at most two. For the entire collection see [Zbl 1493.37005].