Let G = ( V, E ) be a simple graph with vertex setxs V and edge set . A mixed Roman dominating function of is f : ∪ → {0, 1, 2} satisfying the condition that every element x ∈ for which f(x) 0 adjacent or incident to at least one y f(y) 2. The weight ω( ∑ domination number γ R minimum We first show problem computing * NP-complete bipartite graphs then we present upper lower bounds on number, so...

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. called Eisenstein (0, 1)-adjacency We characterize set S for which normal Cayley Cay(Γ,S) HS-integral any finite group Γ. further show that a and only it integral. This paper generalizes results Kadyan Bhattacharjy (2022) [11].

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

We introduce nite relational structures called sketches, that represent edge crossings in drawings of nite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges th...