On Spectral Properties of Lanzhou Matrix of Graphs

Let $\mathrm{\Gamma }$ be a simple graph on $n$ vertices. Lanzhou index is defined as $Lz(\mathrm{\Gamma })=\sum _{u\in V(\mathrm{\Gamma })}{d}_{\mathrm{\Gamma }}(u{)}^2{d}_{\bar{\mathrm{\Gamma }}}(u)\mathrm{.}$ In this manuscript, the Lanzhou matrix, denoted by $A_{Lz}(\mathrm{\Gamma })$, has been defined, and its spectral properties are studied. The $u{v}^{th}$ entry in $A_{Lz}(\mathrm{\Gamma })$ is $d_{\mathrm{\Gamma }}(u)d_{\bar{\mathrm{\Gamma }}}(u)+d_{\mathrm{\Gamma }}(v)d_{\bar{\mathrm{\Gamma }}}(v)$ if $u$ and $v$ are adjacent. Otherwise, the entry is zero. Some bounds on Lanzhou energy and spread on the Lanzhou matrix are obtained. Also, Lanzhou eigenvalues and inertia for some standard graphs have been obtained. Additionally, characterizations for the symmetricity of Lanzhou eigenvalues about the origin are obtained.