A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the ei...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

Given a graph $G$ and an odd prime $p$, for mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ ${\mathbb Z}_p$-boundary $b$ of $G$, orientation $\tau$ is called $(f,b;p)$-orientation if the net out $f$-flow same as $b(v)$ in Z}_p$ at each vertex $v\in V(G)$ under $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations closely related to Tutte's nowhere zero 3...