Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the nec...

We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain certain results for a perturbat...

In this paper, we determine the four trees (resp. the three unicyclic graphs), which share the second to fifth (resp. the second to fourth) largest Laplacian spreads among all the trees (resp. connected unicyclic graphs) on n ≥ 10 (respectively n ≥ 17) vertices.

It was conjectured that for each simple graph G = (V , E) with n = |V (G)| vertices and m = |E(G)| edges, it holdsM2(G)/m ≥ M1(G)/n, whereM1 andM2 are the first and second Zagreb indices. Hansen and Vukičević proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In th...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...

The interpretation of topological resonance energy (TRE), the mathematical measure of energetic chemical aromaticity at the Hückel level of theory, is revisited by providing a concise analysis of the matching polynomial of a chemical graph. Whereas the matching (or acyclic) polynomial is not a properly characteristic polynomial in general, it is the arithmetic mean of the characteristic polynom...

The Randić index, R, also known as the connectivity or branching index, is an important topological index in chemistry. In order to attack some conjectures concerning the Randić index, Dvořák et al. [European J. Combin. 32 (2011), 434–442] introduced a modification of this index, denoted by R′. In this paper we present some of the basic properties of R′. We determine graphs with minimal and max...

• In the second revision of paper we have corrected according to suggestions reviewer. This includes slight two proofs remarks 2. and 6. reviewer, while other are mainly mistypes which accordingly corrected. We thank reviewer for careful reading paper. a graph G , cardinality smallest ordered set vertices that distinguishes every element V ( ) is (vertex) metric dimension . Similarly, such edge...

The energy of a graph $G$ is the sum absolute values eigenvalues adjacency matrix $G$. Let $s^+(G), s^-(G)$ denote squares positive and negative $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if connected order $n$, then $s^+(G)\geq n-1$ $s^-(G) \geq n-1$. In this paper, we show partial results towards conjecture. particular, numerous st...

The firefighter problem is defined as below. A fire initially breaks out at a vertex r on a graph G. In each step, a firefighter chooses to protect one vertex, which is not yet burnt. And the fire spreads out to its unprotected neighboring vertices afterwards. The objective of the problem is to choose a sequence of vertices to protect, in order to save maximum number of vertices from the fire. ...