Lower Bounds on the Eccentric Connectivity Index in Terms of the Irregularity Index

Let G be a connected simple graph of order n≥3, and let ξc(G)=∑v∈V(G)ecG(v)degG(v) denote its eccentric connectivity index. We study the lower bounds for ξc(G) in terms of the irregularity index t(G), defined as the number of distinct vertex degrees of G. For t=t(G)≥2, we prove that ξc(G)≥3n+t2−3t−1, which reduces to the classical bound 3(n−1) when t=2, with equality attained by the star K1,n−1. We also derive refined bounds when the minimum degree δ or maximum degree Δ is prescribed, distinguishing between graphs with and without a universal vertex. The proofs use degree-sequence constraints together with lower estimates for vertex eccentricities. These results clarify how the number of distinct degrees influences the eccentric connectivity index and provide new lower bounds for connected graphs.