A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) to {0, 1, 2}$satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least onevertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independentRoman dominating function (OIRDF) on $G$ if the set ${vin Vmid f(v)=0}$ is independent.The (outer-independent) Roman dom...

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of chromatic function. We introduce vertex-weighted version XB and show that this admits deletion-contraction relation. also demonstrate spanning-tree spanning-forest expansions generalizing those polynomial by connecting to other graph functions. Second, give several methods for constructing non...

The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=∑v∈V(Γ)f(d(v)) for a f(x) on non-negative real numbers. In this paper, we determine the extremal graphs with maximum (minimum) vertex degree in set of all n-vertex chemical trees, and connected graphs. We also present Nordhaus–Gaddum-type results Hf(Γ)+Hf(Γ¯) Hf(Γ)·Hf(Γ¯).

Abstract We study numerically the two-point correlation functions of height in six-vertex model with domain wall boundary conditions. The and are computed by Markov chain Monte-Carlo algorithm. Particular attention is paid to free fermionic point (Δ = 0), for which obtained analytically thermodynamic limit. A good agreement exact numerical results allows us extend calculations disordered (|Δ| &...

‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...