By swapping out atoms for vertices and bonds edges, a graph may be used to model any molecular structure. A G is considered chemical in theory if no vertex of has degree 5 or greater. The bond incident (BID) index defined as the total contributions f(dG(u),dG(v)) from all edges uv G, where dG(w) stands w E(G) set f real-valued symmetric function. This paper addresses problem finding graphs with...

‎Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge‎ ‎set E(G)‎. ‎The (first) edge-hyper Wiener index of the graph G is defined as‎: ‎$$WW_{e}(G)=sum_{{f,g}subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=frac{1}{2}sum_{fin E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$‎ ‎where de(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and de(f|G)=∑g€(G)de(f,g|G). ‎In thi...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

Given a three-dimensional surface G, not necessarily rotationally symmetric, and away from a point source, we design a surface F such that the lens sandwiched between the two surfaces refracts radiation into a given direction or into a given point. The surface F satisfies a system of first-order partial differential equations that can be solved in terms of G and the refractive indices of the me...

Parameters: • Number L of sample slices • Number V feature slices • Number Rrf , Rrs of feature/sample splits • Threshold τ ∈ (0, 1) Input: sample S Init: Ni = 0, i = 1, . . . , d for r = 1 to Rrf do Split feature indices F = {1, . . . , d} randomly into V slices F , . . . , F (V . of equal size b d V c for r = 1 to Rrs do Split the sample S randomly into L slices S, . . . , S of equal size bNL...

Let $mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(Hcap T)H_{G}/H_{G}leq Z_{mathfrak{F}}(G/H_{G})$, where $Z_{mathfrak{F}}(G/H_{G})$ denotes the $mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structur...