For a graph G = (V,E), a Roman dominating function on G is a function f : V (G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by γR (G). T...

In this supplemental article, we provide detailed proofs for the propositions and theorems in the main paper. We write the indicator function of the event A as 1A. Let X1, · · · , Xn and Y1, · · · , Yñ be independent random samples in Rd from unknown distributions F and G, respectively, with corresponding densities f and g with respect to Lebesgue measure. The densities are assumed to be contin...

let $g=(v,e)$ be a connected simple graph. a labeling $f:v to z_2$ induces two edge labelings $f^+, f^*: e to z_2$ defined by $f^+(xy) = f(x)+f(y)$ and $f^*(xy) = f(x)f(y)$ for each $xy in e$. for $i in z_2$, let $v_f(i) = |f^{-1}(i)|$, $e_{f^+}(i) = |(f^{+})^{-1}(i)|$ and $e_{f^*}(i) = |(f^*)^{-1}(i)|$. a labeling $f$ is called friendly if $|v_f(1)-v_f(0)| le 1$. for a friendly labeling $f$ of...

‎let g=(v,e) be a simple connected graph with vertex set v and edge set e. the first, second and third zagreb indices of g are respectivly defined by: $m_1(g)=sum_{uin v} d(u)^2, hspace {.1 cm} m_2(g)=sum_{uvin e} d(u).d(v)$ and $ m_3(g)=sum_{uvin e}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in g and uv is an edge of g connecting the vertices u and v. recently, the first and second m...

(6) For every finite sequence g of elements of R such that r ∈ rngg holds (Inc(g))(1)≤ r and r ≤ (Inc(g))(lenInc(g)). (7) Suppose 1 ≤ i and i ≤ lenh and 1 ≤ I and I ≤ width the Go-board of h. Then (the Go-board of h◦ (1, I))1 ≤ (hi)1 and (hi)1 ≤ (the Go-board of h◦ (len the Go-board of h, I))1. (8) Suppose 1 ≤ i and i ≤ lenh and 1 ≤ I and I ≤ len the Go-board of h. Then (the Go-board of h◦ (I,1...

Let G = (V,E) be a simple graph. An edge labeling f : E → {0, 1} induces a vertex labeling f : V → Z2 defined by f(v) ≡ ∑ uv∈E f(uv) (mod 2) for each v ∈ V , where Z2 = {0, 1} is the additive group of order 2. For i ∈ {0, 1}, let ef (i) = |f−1(i)| and vf (i) = |(f+)−1(i)|. A labeling f is called edge-friendly if |ef (1) − ef (0)| ≤ 1. If (G) = vf (1) − vf (0) is called the edge-friendly index o...