A Degree-Based Spectral Radius Conjecture Refuted: The Second Zagreb Matrix of Unicyclic Graphs

Abstract

A conjecture in [MATCH Commun. Math. Comput. Chem. 89 (2023) 513–530] states that for any unicyclic graph G of order n ≥3, the spectral radius of any degree–based matrix (i.e., a matrix whose entries are functions of vertex degrees) lies between ρ(Cn) and ρ(S+ 3), where S+ 3 denotes the unicyclic graph obtained by attaching n−3 pendent vertices to a single vertex of a triangle. We disprove the upper bound for the second Zagreb matrix. Exhaustive computations for n = 5 to 11 show that the true maximizer is a triangle: for n = 6 it has one leaf on each vertex (1,1,1); for all other n ≥ 5 it has leaf counts (⌊n−3 2 ⌋,⌈n−3 2 ⌉,0) (i.e., one cycle vertex receives no pendent leaf). We conjecture that these graphs uniquely maximize the second Zagreb spectral radius for every n ≥ 5.