The subtree number N(G) of a graph G is defined as the number of non-empty subtrees of G. The Wiener index W(G) of a graph G is defined as the sum of distances over all pairs of vertices in G. It has been noted that, for many families of trees and graphs, the graphs that achieve the largest number of subtrees are exactly those that attain the smallest Wiener index, and vice versa. Consequently, it is often said that the subtree number and the Wiener index have a "negative" correlation. In this paper, we show that, except for extremal graphs, this "negative" correlation does not generally hold. In particular, for every n >= 14, we construct a pair of unicyclic graphs G and H, each having n vertices and identical degree sequences, such that W(G) < W(H) and N(G) < N(H). Furthermore, our construction shows that both differences, N(H) N(G) and W (H) W(G), grow unbounded as n increases.
A matching M in a graph G is a set of pairwise non-adjacent edges. M is maximal if it is not contained in any other matching of G. A minimum maximal matching is a maximal matching of minimum size, and its cardinality is the saturation number of G. Maximal matchings serve as an effective model for characterizing certain real-world problems, while the saturation number captures information about their worst-case scenarios in a certain sense. In this paper, we prove that the saturation number of any phenylene chain with n benzenoids is 2n. We also derive a formula to enumerate minimum maximal matchings via the transfer matrix method. Furthermore, we determine the extremal phenylene chains with n benzenoids achieving the largest and smallest number of minimum maximal matchings, respectively.
This paper gives an affirmative answer to a conjecture proposed by Goldberg (2014) relating to graph energy which says that for a graph G with m edges, 2m2 epsilon(G) >= X p . dxdy xy is an element of E(G)
This paper establishes a comprehensive framework of novel upper bounds for the energy of graphs, rigorously connecting this spectral invariant to a suite of topological indices. We derive a unified collection of theorems that express graph energy in terms of fundamental parameters-order, size, and extreme degrees-while intricately incorporating advanced indices such as the general zeroth-order Randic index, the Sombor index, and the atom-bond connectivity index. Our results not only generalize but also systematically refine many classical bounds in the literature, demonstrating the profound interplay between spectral graph theory and chemical graph theory.
For a graph G = (V-G, E-G), a subset S subset of VG is called a dissociation set if the induced subgraph G[S] does not contain P-3 as a subgraph. A maximum dissociation set of G is a dissociation set with the maximum cardinality and its cardinality is called the dissociation number of G. In this paper, among all trees, bipartite graphs and general connected graphs with fixed order and dissociation number, the sharp lower bounds of the Wiener index are determined and the corresponding extremal graphs are characterized, respectively. Furthermore, the graphs having the maximum Wiener indices with fixed order n and dissociation number phi is an element of {2, r (2)(3) n ],n - 2, n - 1} are also characterized, respectively.
