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...

The vertex (resp. edge) metric dimension of a connected graph G is the size smallest set S?V(G) which distinguishes all pairs vertices edges) in G. In Sedlar and Škrekovski (2021) it was shown that both edge unicyclic always take values from just two explicitly given consecutive integers are derived structure graph. A natural problem arises to determine under what conditions these dimensions ea...

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...