Bounding the largest eigenvalue of signed graphs

In this study we derive certain upper bounds for the largest eigenvalue (called the index and denoted lambda(1)) of a signed graph. In particular, we prove the following upper bound: lambda(2)(1) <= max {d(i)m(i) - n(i) : 1 <= i <= n}, where d(i) is the vertex degree of i, m(i) = 1/d(i) Sigma(j similar to i) d(j) and n(i) = Sigma(j similar to i) (vertical bar N-i(sigma(ij))boolean AND N-j vertical bar-vertical bar vertical bar N-i(sigma(ij))boolean AND N-j(sigma(ij))vertical bar-vertical bar N-i(sigma(ij))boolean AND N-j(-sigma(ij))vertical bar vertical bar), with N-i, N-i(+) and N-i(-) denoting the neighbourhood, the positive neighbourhood and the negative neighbourhood of a vertex i. In our proofs we use standard techniques transferred from the field of (unsigned, simple) graphs. Using the fact that all switching equivalent signed graphs share the same spectrum, we derive some more sophisticated bounds. (C) 2019 Elsevier Inc. All rights reserved.