A signed graph is a graph in which positive or negative signs are assigned to its edges. We consider equitable colouring and Hamiltonian colouring to obtain induced signed graphs. An equitable colour-induced signed graph is a signed graph constructed from a given graph in which each edge uv receives a sign (β1)|c(v)βc(u)|,where c is an equitable colouring of vertex v. A Hamiltonian colour-induced signed graph is a signed graph obtained from a graph G in which for each edge e = uv, the signature function Ο(uv)=(β1)|c(v)βc(u)|, gives a sign such that, |c(u)β c(v)| β₯ n β 1 β D(u, v) where c is a function that assigns a colour to each vertex satisfying the given condition. This paper discusses the properties and characteristics of signed graphs induced by the equitable and Hamiltonian colouring of graphs.
The collection of subgraphs of a graph [Formula: see text] containing [Formula: see text] and the null graph [Formula: see text], which is closed under union and intersection, is said to be a graph topology defined on [Formula: see text]. In this paper, we investigate the idea of spanning graph topology of a graph [Formula: see text], where we consider the collection of spanning subgraphs of [Formula: see text] satisfying the axioms analogous to the axioms of graph topology. We begin with the basic concepts of spanning graph topological space and later and introduce two spanning subgraph complements to define closed graphs in spanning graph topological spaces.
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