Reducing a network by appending strong links to the spanning tree

I am trying to reduce a full graph so that it only shows the strongest links.

To do that I am doing this:

1. Multiply the original matrix by -1 (the values are in the interval [0,1])
2. Obtain the minimum spanning tree, step 1. makes it a spanning tree with the strongest links
3. Paste other strong links that are not a part of the spanning tree until the average number of links per node is <= 5.

My function for this goal is:

# number of links
n <- 10 
n2 <- sqrt(n)

library(igraph)

set.seed(1234)
mat <- matrix(rnorm(n, 1, 0.5), nrow = n2, ncol = n2)
mat[mat < 0] <- 0
colnames(mat) <- letters[1:n2]
rownames(mat) <- letters[1:n2]
diag(mat) <- 1

trim_network_1 <- function(mat, avg_links = 5, tolerance = 0.01) {
  mat <- (-1) * mat

  g <- graph_from_adjacency_matrix(mat, weighted = TRUE, mode = "undirected", diag = FALSE)

  g_mst <- mst(g, algorithm = "prim")

  threshold <- 0
  avg_links_n <- FALSE

  while (avg_links_n == FALSE) {
    if (threshold < 1) {
      message(sprintf("%s threshold...", threshold))

      g_not_in_mst <- delete.edges(g, which(abs(E(g)$weight) <= threshold))
      g_not_in_mst <- graph.difference(g_not_in_mst, g_mst)

      g <- rbind(
        as_data_frame(g_mst),
        as_data_frame(g_not_in_mst)
      )

      g <- graph_from_data_frame(g, directed = F)

      avg_links_n <- ifelse(mean(degree(g)) <= avg_links, TRUE, FALSE)
      threshold <- threshold + tolerance

      if (avg_links_n == TRUE) {
        message(sprintf("%s threshold achieves the avg number of connections", threshold))
        E(g)$weight <- (-1) * E(g)$weight
        return(g)
      }
    } else {
      warning("no threshold achieves the avg number of connections\nreturning maximum spanning tree")
      avg_links_n <- TRUE
      E(g_mst)$weight <- (-1) * E(g_mst)$weight
      return(g_mst)
    }
  }
}

trim_network_2 <- function(mat, avg_links = 5, tolerance = 0.01) {
  mat <- (-1) * mat

  g <- graph_from_adjacency_matrix(mat, weighted = TRUE, mode = "undirected", diag = FALSE)

  g_mst <- mst(g, algorithm = "prim")

  threshold <- 0
  avg_links_n <- FALSE

  while (avg_links_n == FALSE) {
    if (threshold < 1) {
      message(sprintf("%s threshold...", threshold))

      g_not_in_mst <- delete.edges(g, which(abs(E(g)$weight) <= threshold))
      g_not_in_mst <- graph.difference(g_not_in_mst, g_mst)

      g <- graph.union(g_mst, g_not_in_mst)
      E(g)$weight <- pmin(E(g)$weight_1, E(g)$weight_2, na.rm = T)
      g <- remove.edge.attribute(g, "weight_1")
      g <- remove.edge.attribute(g, "weight_2")

      avg_links_n <- ifelse(mean(degree(g)) <= avg_links, TRUE, FALSE)
      threshold <- threshold + tolerance

      if (avg_links_n == TRUE) {
        message(sprintf("%s threshold achieves the avg number of connections", threshold))
        E(g)$weight <- (-1) * E(g)$weight
        return(g)
      }
    } else {
      warning("no threshold achieves the avg number of connections\nreturning maximum spanning tree")
      avg_links_n <- TRUE
      E(g_mst)$weight <- (-1) * E(g_mst)$weight
      return(g_mst)
    }
  }
}

g1 <- trim_network_1(mat)
g2 <- trim_network_2(mat)

g1 <- as_data_frame(g1)
g2 <- as_data_frame(g2)

g1w <- order(g1$weight)
g1 <- g1[g1w, ]

g2w <- order(g2$weight)
g2 <- g2[g2w, ]

# this is symmetric so sometimes A-B is returned as B-A
# all.equal(g1, g2) doens't always hold bc of symmetry
all.equal(g1$weight, g2$weight)
all.equal(nrow(g1), nrow(g2))

I tried with graph.union(g_mst, g_not_in_mst) and even after doing E(g)$weight <- pmin(E(g)$weight_1, E(g)$weight_2, na.rm = T) I cannot obtain the same result as with the rbind() part.

Any ideas? an "ideal" result would be not to convert to data frame to then convert to graph again.