paper

Arithmetic-Geometric Spectral Radius of Trees and Unicyclic Graphs

arXiv:2105.03884

Abstract

The arithmetic-geometric matrix $A_{a g} (G)$ of a graph $G$ is a square matrix, where the $(i, j)$ -entry is equal to $\frac{d _{i} + d _{j}}{2 d _{i} d _{j}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$ , denoted by $I ¨ _{a g} (G)$ , is the largest eigenvalue of the arithmetic-geometric matrix $A_{a g} (G)$ . Let $S_{n}$ be the star of order $n \geq 3$ and $S_{n} + e$ be the unicyclic graph obtained from $S_{n}$ by adding an edge. In this paper, we prove that for any tree $T$ of order $n \geq 2$ , $2 cos \frac{I ¨}{} n + 1 \leq I ¨ _{a g} (P_{n}) \leq I ¨ _{a g} (T) \leq I ¨ _{a g} (S_{n}) = \frac{n}{2},$ with equality if and only if $T ≅ P_{n}$ for the lower bound, and if and only if $T ≅ S_{n}$ for the upper bound. We also prove that for any unicyclic graph $G$ of order $n \geq 3$ , $2 = I ¨ _{a g} (C_{n}) \leq I ¨ _{a g} (G) \leq I ¨ _{a g} (S_{n} + e),$ the lower (upper, respectively) bound is attained if and only if $T ≅ C_{n}$ ( $T ≅ S_{n} + e$ , respectively) and $I ¨ _{a g} (S_{n} + e) < \frac{n}{2}$ for $n \geq 7$ .

26 pages, 64 figures