sweep_transition_lambda2_min.m

function lambda2_min=sweep_transition_lambda2_min(n,finite_omega_values,v)

    steps=size(finite_omega_values, 2);

    % only the directed chain graph is tested
    L_full=-eye(n);
    for i=1:n-1
        L_full(i,i+1)=1;
    end
    L=L_full(1:n-1,:)*(-1);
    valid_graphs(:,:,1)=L;
    %for testing only
    % valid_graphs(:,:,1) = [1,-1,0,0,0;
    %                         0,1,-1,0,0;
    %                         0,0,1,-1,0;
    %                         0,0,0,1,-1];
    % valid_graphs(:,:,1) = [1,0,-1,0,0;
    %                         0,1,-1,0,0;
    %                         -1,-1,3,-1,0;
    %                         0,0,-1,2,-1];
    % valid_graphs(:,:,1) = [1,0,0,0,-1;
    %                         0,1,0,0,-1;
    %                         0,0,1,0,-1;
    %                         0,0,0,1,-1];
    num_valid_graph=1;

    Omg_bare_temp=eye(n,n);    %define Omega(k)
    for j=1:n-1
        Omg_bare_temp(j+1,j)=-1;
    end
    Omg_bare=Omg_bare_temp(:,1:n-1);        

    omega_matrix=zeros(n-1,n-1);
    for counter=1:(steps)^(n-1)
        quotient=counter;
        for j=1:n-1
            bitmask=rem(quotient,steps);
            quotient=fix(quotient/steps);
            omega_matrix(j,j)=finite_omega_values(bitmask+1);
        end
        omega_matrix_big(:,:,counter)=omega_matrix;
    end  
    num_workload_dist=counter;

    I_matrix=zeros(n,n-1);
    I_matrix2=zeros(n,n-1);
    counter_total=2^(n-1);
    for counter=1:counter_total
        quotient=counter;
        for j=1:n-1
            bitmask=rem(quotient,2);
            quotient=fix(quotient/2);
            if bitmask==1
                I_matrix(j,j)=v;
                I_matrix2(j+1,j)=v;
            else
                I_matrix(j,j)=-v;
                I_matrix2(j+1,j)=-v;
            end
        end
        I_matrix_big(:,:,counter)=I_matrix;
        I_matrix_big(:,:,counter_total+counter)=I_matrix2;
    end
    num_I_matrix=counter_total+counter;

    counter_lambda2_not_larger_zero=0;
    lambda2_min=10;
    lambda2_max_vec=ones(num_workload_dist,1)*0;
    lambda2_max_vec_index=ones(num_workload_dist,1);

    good_graph_counter=0;
    for i=1:num_valid_graph
        this_graph_is_good=1;
        for j=1:num_workload_dist
            %graph_matrix(:,:)=valid_graphs(i,:,:);
            %omega_matrix=omega_matrix_big(j,:,:);
            for k=1:num_I_matrix
                Omega=(Omg_bare*omega_matrix_big(:,:,j)+I_matrix_big(:,:,k))*valid_graphs(:,:,i);
                eigen_omega=sort(eig(0.5*Omega.'+0.5*Omega));
                lambda2_temp=eigen_omega(2);
                if lambda2_temp<lambda2_min
                    lambda2_min=lambda2_temp;
                    lambda2_min_index=[i,j];
                end
                if lambda2_temp>lambda2_max_vec(j)
                    lambda2_max_vec(j)=lambda2_temp;
                    lambda2_max_vec_index(j)=i;
                end
                if lambda2_temp<=0&&isreal(lambda2_temp)
                    counter_lambda2_not_larger_zero=counter_lambda2_not_larger_zero+1;
                    lambda2_index(:,:,counter_lambda2_not_larger_zero)=[i,j];
                    this_graph_is_good=0;

                    return
                end
            end
        end
    end
end
% a_graph=     [1     0    -1     0     0;
%            0     1    -1     0     0;
%            0     0     1    -1     0;
%            0     0     0     1    -1];
% syms ome1 ome2 ome3 ome4 ome5 ome6 ome7 ome8 ome9
%        omega_vector=[ome1 0 0 0
%            0 ome2 0 0
%            0 0 ome3 0
%            0 0 0 ome4];
%        omega_vector=[10 0 0 0
%            0 1 0 0
%            0 0 3 0
%            0 0 0 2];
%       0.5*Omg_bare* omega_vector*a_graph+0.5*a_graph'* omega_vector*Omg_bare'