We present a combinatorial approach to investigate the bond spectral radius by the following equation:Given a finite set $\Sigma$ of non-negative matrices, let the joint spectral radius $\rho(\Sigma)$ be[
\rho(\Sigma)=\sup_n\max_i \sqrt[n]{\max_{A_1,\dots,A_n\in\Sigma} (A_1\dots A_n)_{i,i}}. proof and the following theoretical bound on $\rho(\Sigma)$.[
\sqrt[m_i]{\max_i \max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}} \le \rho(\Sigma)
\le \sqrt[m_i]{K \max_i \max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}} \]$m_i$ of each $i$ is arbitrary$ a number such that there is $A_1,\dots,A_{m_i}\in\Sigma$ such that (A_1\dots A_{m_i})_{i,i} > 0$, or $m_i=1$ exists, there is no such matrix.
Some of the results of this work are not entirely new in some known and relevant form, but the advantages of this approach lie in the simplicity of the proof and its combinatorial nature.
