For ??[0,1], let A?(G?)=?D(G)+(1??)A(G?), where G is a simple undirected graph, D(G) the diagonal matrix of its vertex degrees and A(G?) adjacency signed graph G? whose underlying G. In this paper, basic properties A?(G?) are obtained, positive semidefiniteness studied some bounds on eigenvalues derived—in particular, lower upper largest eigenvalue obtained.

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study graphs, we show that notion 4-coloring graphs and captured, through simple operations, by C−4. In particular, 4-color theorem equivalent to: Given a planar G, obtained from G replacing edge with path length 2 maps W...

The computer code developed previously (K. Balasubramanian, J . Computational Chern., 5,387 (1984)) for the characteristic polynomials of ordinary (nonweighted) graphs is extended in this investigation to edge-weighted graphs, heterographs (vertex-weighted), graphs with loops, directed graphs, and signed graphs. This extension leads to a number of important applications of this code to several ...