‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

We present a quantum mechanical model for four-wave mixing Josephson traveling-wave parametric amplifier including substrate losses and associated thermal fluctuations. Under the assumption of strong undepleted classical pump tone, we derive an analytic solution bosonic annihilation operator weak signal photon field using temporal equations motion in reference timeframe, chromatic dispersion. F...

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...

Fuzzy Water Quality Index (FWQI) was applied in order to assess the degree of drinking water resources in Yazd province, Iran. Thisstudy has also offered the creation of a new fuzzy water quality index (FWQI) to evaluate this tool’s applicability. 12 chemicalparameters including toxic and non-toxic heavy metals measured in 71 groundwater samples collected from drinking water resourcesin rural a...

A snark is a cyclically-4-edge-connected cubic graph with chromatic index 4. In 1880, Tait proved that the Four-Color Conjecture is equivalent to the statement that every planar bridgeless cubic graph has chromatic index 3. The search for counter-examples to the FourColor Conjecture motivated the definition of the snarks. A k-total-coloring of G is an assignment of k colors to the edges and ver...

The adaptable chromatic number of a graph G is the smallest integer k such that for any edge k-colouring of G there exists a vertex kcolouring of G in which the same colour never appears on an edge and both its endpoints. (Neither the edge nor the vertex colourings are necessarily proper in the usual sense.) We give an efficient characterization of graphs with adaptable chromatic number at most...

A bound for the (n,m)-mixed chromatic number in terms of the chromatic number of the square of the underlying undirected graph is given. A similar bound holds when the chromatic number of the square is replaced by the injective chromatic number. When restricted to n = 1 andm = 0 (i.e., oriented graphs) this gives a new bound for the oriented chromatic number. In this case, a slightly improved b...