$A$ is an [adjacent matrix](http://en.wikipedia.org/wiki/Adjacency_matrix) of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector.


I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all real constants, $bA+c$ is obtained by multiply each element of $A$ by $b$ and then increased it by $c$, $I$ is the Identity matrix)


Let $lac$ be the largest eigenvalue of $bA+c-dI$.


I want to obtain an approximate function: $lac=F(la,c,..)$. Could you help me on this?


Thanks.



Eigenvalues of sum of an adjacent matrix and a constant
An easy upper bound is $la+nc$, where $n$ is the dimension. This is because the matrix norm of a sum is no more than the sum of the matrix norms.


An easy lower bound is $k+nc$, where $k$ is the average vertex degree. This is because, for all $v$, $v \cdot Mv/||v||^2$ is no more than the highest eigenvalue of $M$. The vector of all $1$s provides this lower bound.


If your graph is approximately regular, $la$ is not much more than $k$, and these bounds will be relatively close.


