The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs

In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number \(\varepsilon <\frac{1}{3}\) $\epsilon <\frac{1}{3}$ , if n is a sufficiently large integer, then any graph G of order n with \(\rho (G)>\sqrt{\left\lfloor \frac{n^{2}}{4}\right\rfloor }\) $\rho (G)>\sqrt{⌊\frac{n^2}{4}⌋}$ contains a cycle of length t for all integers \(t\in [3,(\frac{1}{3}-\varepsilon )n]\) $t\in [3,(\frac{1}{3}-\epsilon )n]$ , where \(\rho (G)\) $\rho (G)$ is the spectral radius of G. This improves the result of Li and Ning (2023).