diff --git a/src/graph.jl b/src/graph.jl
index 948ac32e..23331789 100644
--- a/src/graph.jl
+++ b/src/graph.jl
@@ -270,13 +270,9 @@ end
 degree(g::AdjacencyGraph{T,false}, v::Integer) where {T} = g.S.colptr[v + 1] - g.S.colptr[v]
 
 function degree(g::AdjacencyGraph{T,true}, v::Integer) where {T}
-    d = 0
-    for u in neighbors(g, v)
-        if u != v
-            d += 1
-        end
-    end
-    return d
+    neigh = neighbors(g, v)
+    has_selfloop = insorted(v, neigh)
+    return g.S.colptr[v + 1] - g.S.colptr[v] - has_selfloop
 end
 
 nb_edges(g::AdjacencyGraph{T,false}) where {T} = nnz(g.S) ÷ 2
diff --git a/src/order.jl b/src/order.jl
index 9cefc1b2..76491d39 100644
--- a/src/order.jl
+++ b/src/order.jl
@@ -80,17 +80,17 @@ Instance of [`AbstractOrder`](@ref) which sorts vertices using their degree in t
 struct LargestFirst <: AbstractOrder end
 
 function vertices(g::AdjacencyGraph, ::LargestFirst)
-    criterion(v) = degree(g, v)
+    degrees = map(Base.Fix1(degree, g), vertices(g))
+    criterion(v) = degrees[v]
     return sort(vertices(g); by=criterion, rev=true)
 end
 
 function vertices(bg::BipartiteGraph{T}, ::Val{side}, ::LargestFirst) where {T,side}
     other_side = 3 - side
     n = nb_vertices(bg, Val(side))
-    visited = falses(n)  # necessary for distance-2 neighborhoods
+    visited = fill(false, n)  # necessary for distance-2 neighborhoods
     degrees_dist2 = zeros(T, n)
     for v in vertices(bg, Val(side))
-        fill!(visited, false)
         for u in neighbors(bg, Val(side), v)
             for w in neighbors(bg, Val(other_side), u)
                 if w != v && !visited[w]
@@ -99,6 +99,11 @@ function vertices(bg::BipartiteGraph{T}, ::Val{side}, ::LargestFirst) where {T,s
                 end
             end
         end
+        for u in neighbors(bg, Val(side), v)
+            for w in neighbors(bg, Val(other_side), u)
+                visited[w] = false  # reset only the toggled ones to false
+            end
+        end
     end
     criterion(v) = degrees_dist2[v]
     return sort(vertices(bg, Val(side)); by=criterion, rev=true)
@@ -330,13 +335,14 @@ end
 function vertices(
     g::AdjacencyGraph{T}, order::DynamicDegreeBasedOrder{degtype,direction}
 ) where {T<:Integer,degtype,direction}
+    true_degrees = degrees = T[degree(g, v) for v in vertices(g)]
     if degree_increasing(; degtype, direction)
         degrees = zeros(T, nb_vertices(g))
     else
-        degrees = T[degree(g, v) for v in vertices(g)]
+        degrees = true_degrees
     end
     db = DegreeBuckets(
-        T, degrees, maximum_degree(g); reproduce_colpack=order.reproduce_colpack
+        T, degrees, maximum(true_degrees); reproduce_colpack=order.reproduce_colpack
     )
     π = T[]
     sizehint!(π, nb_vertices(g))
@@ -359,15 +365,21 @@ function vertices(
     # compute dist-2 degrees in an optimized way
     n = nb_vertices(g, Val(side))
     degrees_dist2 = zeros(T, n)
-    dist2_neighbor = falses(n)
+    visited = fill(false, n)
     for v in vertices(g, Val(side))
-        fill!(dist2_neighbor, false)
         for w1 in neighbors(g, Val(side), v)
             for w2 in neighbors(g, Val(other_side), w1)
-                dist2_neighbor[w2] = true
+                if w2 != v && !visited[w2]
+                    degrees_dist2[v] += 1
+                    visited[w2] = true
+                end
+            end
+        end
+        for w1 in neighbors(g, Val(side), v)
+            for w2 in neighbors(g, Val(other_side), w1)
+                visited[w2] = false
             end
         end
-        degrees_dist2[v] = sum(dist2_neighbor)
     end
     if degree_increasing(; degtype, direction)
         degrees = zeros(T, n)
@@ -378,7 +390,7 @@ function vertices(
     db = DegreeBuckets(T, degrees, maxd2; reproduce_colpack=order.reproduce_colpack)
     π = T[]
     sizehint!(π, n)
-    visited = falses(n)
+    fill!(visited, false)
     for _ in 1:nb_vertices(g, Val(side))
         u = pop_next_candidate!(db; direction)
         direction == :low2high ? push!(π, u) : pushfirst!(π, u)
@@ -393,7 +405,11 @@ function vertices(
                 update_bucket!(db, v; degtype, direction)
             end
         end
-        fill!(visited, false)
+        for w in neighbors(g, Val(side), u)
+            for v in neighbors(g, Val(other_side), w)
+                visited[v] = false  # reset only the toggled ones to false
+            end
+        end
     end
     return π
 end
diff --git a/test/order.jl b/test/order.jl
index 594d3be8..16b8fde2 100644
--- a/test/order.jl
+++ b/test/order.jl
@@ -63,12 +63,13 @@ end;
 @testset "LargestFirst" begin
     for has_diagonal in (false, true)
         A = sparse([
-            0 1 0
-            1 0 1
-            0 1 0
+            0 1 0 0
+            1 0 1 1
+            0 1 0 1
+            0 1 1 0
         ])
         ag = AdjacencyGraph(A; has_diagonal)
-        @test vertices(ag, LargestFirst()) == [2, 1, 3]
+        @test vertices(ag, LargestFirst()) == [2, 3, 4, 1]
     end
 
     A = sparse([