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The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$ terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of $k=2$, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number $n$ of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) $n \geq 3$ and $k$ is odd, (b) $n=1$, or (c) $n=2$ and $k \equiv 1 \pmod{6}$. Two proofs are presented of the $n=2$ case -- the other situations were handled previously -- and we use the argument further to show that the distance spectral radius for $n=2$ is equal to $2^{k-1}-1$. Some related open questions are also discussed., Comment: 6 pages, 0 figures

The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called $D_z$- and $D_{z+1}$-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.

Three general molecular descriptors, namely the general sum-connectivity index, general Platt index and ordinary generalized geometric-arithmetic index, are studied here. Best possible bounds for the aforementioned descriptors of arbitrary saturated hydrocarbons are derived. These bounds are expressed in terms of number of carbon atoms and number of carbon-carbon bonds of the considered hydrocarbons., Comment: 21 pages and 3 figures

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph $G$, the ABC index is defined as $\sum_{uv\in E}\sqrt{\frac{d(u) +d(v)-2}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ denotes the set of edges of $G$. In this paper we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a $B_4$-branch and $B_1$ or $B_2$-branches.

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. It found considerable applications in a variety of fields. In this paper, we determine the minimum Kirchhoff index among the unicyclic graphs with fixed number of vertices and matching number, and characterize the extremal graphs.

We obtain the maximum sum-connectivity indices of graphs in the set of trees and in the set of unicyclic graphs respectively with given number of vertices and maximum degree, and determine the corresponding extremal graphs. Additionally, we deduce the n-vertex unicyclic graphs with the first two maximum sum-connectivity indices for $n\ge 4$., Comment: 12 pages

The reverse degree distance is a connected graph invariant closely related to the degree distance proposed in mathematical chemistry. We determine the unicyclic graphs of given girth, number of pendant vertices and maximum degree, respectively, with maximum reverse degree distances.

The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity index, and the n-vertex trees of pendent vertices, with the maximum eccentric connectivity index. We also determine the n-vertex trees with respectively the minimum, second-minimum and third-minimum, and the maximum, second-maximum and third-maximum eccentric connectivity indices for, Comment: 18 pages, 2 figures

We determine the minimum sum--connectivity index of bicyclic graphs with $n$ vertices and matching number $m$, where $2\le m\le \lfloor\frac{n}{2}\rfloor$, the minimum and the second minimum, as well as the maximum and the second maximum sum--connectivity indices of bicyclic graphs with $n\ge 5$ vertices. The extremal graphs are characterized.