The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.
The epidemic modeling is recognized as an effective approach to the understanding of propagation process of objects over a network [
Hypercubes are a class of regular networks [
This paper addresses three subproblems of the SRMP, where two/three/four edges are removed from a hypercube, respectively. First, for each of the three subproblems of the SRMP, a candidate optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.
The remaining materials are organized in this fashion: the preliminary knowledge is given in Section
For fundamental knowledge on the spectral radius of a network, see [
An
Three examples of
An
This section considers the optimal scheme of deleting two/three/four edges from
Firstly, we consider a subproblem of the SRMP, denoted by SRMP-H2, for which two edges will be deleted from a hypercube. Let us present a candidate for the optimal solution to the SRMP-H2 as follows, where
Figure
The proposed candidate in
For
Assume that the red edge in the upper-left 3D subcube of
For
The proposed candidate (red) versus
Therefore, we propose the following conjecture.
For all
Secondly, we consider a subproblem of the SRMP problem, denoted by SRMP-H3, for which three edges will be removed from a hypercube. Let us present a candidate for the optimal solution to the SRMP-H3 as follows, where
The proposed candidate in
For
The proposed candidate (red) versus
For all
Finally, consider a subproblem of the SRMP, denoted by SRMP-H4, for which four edges will be deleted from a hypercube. Let us present a candidate to the optimal solution to the SRMP-H4 as follows, where
If
If
Figure
The proposed candidate in
For
The proposed candidate (red) versus
For all
This paper has addressed the spectral radius minimization problem for hypercubes. Given the number of edges to be deleted, a candidate for the optimal solution has been presented. For small-sized hypercubes, the proposed candidate has been shown to be optimal. For medium-sized hypercubes, it has been shown that the proposed candidate is likely to be optimal. Due to the symmetry of hypercubes, there are multiple optimal solutions for each of the subproblems. The experimental results show that, up to isomorphism, all of the optimal solutions are identical. By observing the pattern of the proposed candidate, it has been speculated that, for any symmetric network, the edges in an optimal solution are always evenly distributed.
Towards this direction, some researches are yet to be done. First, the proposed conjectures need a proof. Second, this work should be extended to asymmetric networks such as the hypercube-like networks [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by Science and Technology Support Program of China (Grant no. 2015BAF05B03), Natural Science Foundation of China (Grants nos. 61572006, 71301177), Basic and Advanced Research Program of Chongqing (Grant no. cstc2013jcyjA1658), and Fundamental Research Funds for the Central Universities (Grant no. 106112014CDJZR008823).