On the Estrada index of unicyclic and bicyclic signed graphs

2023/09/23 by Shamsher, Tahir, Pirzada, S., Bhat, Mushtaq A.
#05C22 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.13252

Abstract

Let Γ=(G, σ) be a signed graph of order n with eigenvalues μ₁,μ₂,…,μ_n. We define the Estrada index of a signed graph Γ as EE(Γ)=∑_i=1ⁿe^μ_i. We characterize the signed unicyclic graphs with the maximum Estrada index. The signed graph Γ is said to have the pairing property if μ is an eigenvalue whenever -μ is an eigenvalue of Γ and both μ and -μ have the same multiplicities. If Γ_p^-(n, m) denotes the set of all unbalanced graphs on n vertices and m edges with the pairing property, we determine the signed graphs having the maximum Estrada index in Γ_p^-(n, m), when m=n and m=n+1. Finally, we find the signed graphs among all unbalanced complete bipartite signed graphs having the maximum Estrada index.

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