Milovan Ilic

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertices of degree $k$. For $k = 1$, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order $n$.

Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed graph invariant, defined as the sum of square roots of Laplacian eigenvalues. For bipartite graphs, the Laplacian--like energy coincides with the recently defined incidence energy $IE (G)$ of a graph. In [D. Stevanovi\' c, \textit{Laplacian--like energy of trees}, MATCH Commun. Math. Comput. Chem. 61 (2009), 407--417.] the author introduced a partial ordering of graphs based on Laplacian coefficients. We point out that original proof was incorrect and illustrate the error on the example using Laplacian Estrada index. Furthermore, we found the inverse of Jacobian matrix with elements representing derivatives of symmetric polynomials of order $n$, and provide a corrected elementary proof of the fact: Let $G$ and $H$ be two $n$-vertex graphs; if for Laplacian coefficients holds $c_k (G) \leqslant c_k (H)$ for $k = 1, 2, ..., n - 1$, then $LEL (G) \leqslant LEL (H)$. In addition, we generalize this theorem and provide a necessary condition for functions that satisfy partial ordering based on Laplacian coefficients.