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. ...

A locally irregular graph is a in which the end vertices of every edge have distinct degrees. coloring G any such that each colors induces subgraph G. colorable if it allows coloring. The chromatic index G, denoted by χirr′(G), smallest number used local irregularity conjecture claims all graphs, except odd-length paths, cycles and certain class cacti are three colors. As valid for graphs with ...

A set <em>W</em> is called a local resolving of <em>G</em> if the distance <em>u</em> and <em>v</em> to some elements are distinct for every two adjacent vertices in <em>G</em>. The metric dimension minimum cardinality connected graph split <em>V</em>(<em>G</em>) can be partitioned into subsets <em>V</e...

Given a set ξ = {H1,H2, · · ·} of connected non acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Ŵk,ξ be theexponential generating function (EGF for brief) of connected ξ-free graphs of excess equal to k (k ≥ 1). For each fixed ξ, a fundamental differen...

Given a set ξ = {H1,H2, · · ·} of connected non acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Ŵk,ξ be theexponential generating function (EGF for brief) of connected ξ-free graphs of excess equal to k (k ≥ 1). For each fixed ξ, a fundamental differen...