A Function on Bounds of the Spectral Radius of Graphs

Abstract

Let $G=(V,E)$ be a simple connected graph with $7$ vertices. The degree of $v_i\in V$ and the average of degrees of the vertices adjacent to $v_i$ are denoted by $d_i$ and $m_i$, respectively. The spectral radius of $G$ is denoted by $\rho (G)$. In this paper, we introduce a parameter into an equation of adjacency matrix, and obtain two inequalities for upper and lower bounds of spectral radius. By assigning different values to this parameter, one can obtain some new and existing results on spectral radius. Specially, if $G$ is a nonregular graph, then

$$\rho (G)\le \underset{1\le j<i\le n}{max}\{\frac{d_i{m}_i–d_j{m}_j+\sqrt{(d_i{m}_i–d_j{m}_j{)}^2–4d_i{d}_j(d_i-d_j)(m_i–m_j)}}{2(d_i-d_j)}\}$$ and $$\rho (G)\ge \underset{1\le j<i\le n}{min}\{\frac{d_i{m}_i–d_j{m}_j+\sqrt{(d_i{m}_i–d_j{m}_j{)}^2–4d_i{d}_j(d_i-d_j)(m_i–m_j)}}{2(d_i-d_j)}\}.$$ If $G$ is a bidegreed graph whose vertices of same degree have equal average of degrees, then the equality holds.