2023/05/04 by Shamsher, Tahir, Rashid, Mir Riyaz ul, Pirzada, S.
#05C22 #05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.02964
A signed graph S=(G, σ) is a pair in which G is an underlying graph and σ is a function from the edge set to \±1\. For signed graphs S1 and S2 on n1 and n2 vertices, respectively, the signed neighbourhood corona S1 ⋆s S2 (in short s-neighbourhood corona) of S1 and S2 is the signed graph obtained by taking one copy of S1 and n1 copies of S2 and joining every neighbour of the ith vertex of S1 with the same sign as the sign of incident edge to every vertex in the ith copy of S2. In this paper, we investigate the adjacency, Laplacian and net Laplacian spectrum of S1 ⋆s S2 in terms of the corresponding spectrum of S1 and S2. We determine (i) the adjacency spectrum of S1 ⋆s S2 for arbitrary S1 and net regular S2, (ii) the Laplacian spectrum for regular S1 and regular and net regular S2 and (iii) the net Laplacian spectrum for net regular S1 and arbitrary S2. As a consequence, we obtain the signed graphs with 4 and 5 distinct adjacency, Laplacian and net Laplacian eigenvalues. Finally, we show that the signed neighbourhood corona of two signed graphs is not determined by its adjacency (resp., Laplacian, net Laplacian) spectrum.