Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

Let $D$ be a weighted oriented graph and $I(D)$ its edge ideal. If contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary sufficient for the equality ordinary symbolic powers ideal having each in some it. characterize naturally unicyclic graphs with unique cycles even their ideals. D^{\prime} obtained from afte...

A cactus graph is a connected graph in which any two cycles have at most one vertex in common. Let γ(G) and γc(G) be the domination number and connected domination number of a graph G, respectively. We can see that γ(G) ≤ γc(G) for any graph G. S. Arumugam and J. Paulraj Joseph [1] have characterized trees, unicyclic graphs and cubic graphs with equal domination and connected domination numbers...

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for theMerrifield–Simmons indices of u...

Let L be subset of {3,4,…} and let Xn,M(L) the number cycles belonging to unicyclic components whose length is in random graph G(n,M). We find limiting distribution subcritical regime M=cn with c<1/2 critical M=n2(1+μn−1/3) μ=O(1). Depending on a condition involving series ∑ℓ∈Lzℓ/(2ℓ), we obtain limit either Poisson or normal as n→∞.

Let $d_G(v)$ be the degree of vertex $v$ in a graph $G$. The Sombor index $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which new degree-based topological introduced by Gutman. $\mathscr{T}_{n,\Delta}$ and $\mathscr{U}_{n,\Delta}$ set trees unicyclic graphs with $n$ vertices maximum $\Delta$, respectively. In this paper, tree minimum among are characterized.

In this paper, we obtain the upper and lower bounds on the eccen- tricity connectivity index of unicyclic graphs with perfect matchings. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given matching numbers.

The degree-based entropy Id(G) of a graph G on m>0 edges is obtained from the well-known Shannon −∑i=1np(xi)logp(xi) in information theory by replacing probabilities p(xi) fractions dG(vi)2m, where {v1,v2,…,vn} vertex set G, and dG(vi) degree vi. We continue earlier work Id(G). Our main results deal with effect number operations value also illustrate relevance these applying some to prove extre...

In a graph G, vertex (resp. an edge) metric generator is set of vertices S such that any pair edges) from G distinguished by at least one S. The cardinality smallest the dimension G. Sedlar and Škrekovski (0000) we determined unicyclic graphs it takes its value two consecutive integers. Therein, several cycle configurations were introduced greater values only if these present in graph. this pap...