Let G(V,X) be a finite and simple graph of order n and size m. The complement of G, denoted by G¯, is the graph obtained by removing the lines of G and adding the lines that are not in G. A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we define δ-complement and δ′-complement of a graph as follows. For any two points u and v of G with degu=degv remove the lines between u and v in G and add the lines between u and v which are not in G. The graph thus obtained is called δ-complement of G. For any two points u and v of G with degu≠degv remove the lines between u and v in G and add the lines between u and v that are not in G. The graph thus obtained is called δ′-complement of G. The graph G is δ(δ′)-self-complementary if G≅Gδ(G≅Gδ′). The graph G is δ(δ′)-co-self-complementary if Gδ≅G¯(Gδ′≅G¯). This paper presents different properties of δ and δ′-complement of a given graph.
Let [Formula: see text] be a partition of vertex set [Formula: see text] of order [Formula: see text] of a graph [Formula: see text]. The [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for all [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text], remove the edges between [Formula: see text] and [Formula: see text] in [Formula: see text] and add the edges between [Formula: see text] and [Formula: see text] which are not in [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text]. For a graph [Formula: see text], [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for each [Formula: see text] remove the edges of [Formula: see text] inside [Formula: see text] and add the edges of [Formula: see text] by joining the vertices of [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text] for some partition [Formula: see text] of order [Formula: see text]. In this paper, we determine generalized self-complementary graphs of forest, double star and unicyclic graphs.
Let [Formula: see text] be the adjacency matrix of a graph [Formula: see text]. Let [Formula: see text] denote the row entries of [Formula: see text] corresponding to the vertex [Formula: see text] of [Formula: see text]. The Hamming distance between the strings [Formula: see text] and [Formula: see text] is the number of positions in which [Formula: see text] and [Formula: see text] differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of [Formula: see text].
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