Ordering of minimal energies in unicyclic signed graphs

Abstract

Let S = (G, σ) be a signed graph of order n and size m and let t 1 , t 2 , . . ., t n be the eigenvalues of S. The energy of S is defined as E(S) = n j=1 |t j |.A connected signed graph is said to be unicyclic if its order and size are same.In this paper, we characterize, up to switching, the unicyclic signed graphs with first 11 minimal energies for all n ≥ 12.For 3 ≤ n ≤ 7, we provide complete ordering of unicyclic signed graphs with respect to energy.For n = 8, 9, 10 and 11, we determine unicyclic signed graphs with first 13 minimal energies respectively.