diff --git a/DESCRIPTION b/DESCRIPTION
index 65eb683..0b26088 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -11,8 +11,8 @@ Authors@R: c(
 Description: Cross-validated eigenvalues are estimated by
     splitting a graph into two parts, the training and the test graph.
     The training graph is use to estimate eigenvectors, and
-    the test graph is use to evaluate the correlation between the sample
-    eigenvectors and the eigenspace of the test graph.
+    the test graph is use to evaluate the correlation between the training
+    eigenvectors and the eigenvectors of the test graph.
     The correlations follow a simple central limit theorem that can
     be used to estimate graph dimension via hypothesis testing.
 License: GPL (>= 3)
diff --git a/NAMESPACE b/NAMESPACE
index bed5cff..5b197cc 100644
--- a/NAMESPACE
+++ b/NAMESPACE
@@ -2,23 +2,7 @@
 
 S3method(plot,eigcv)
 S3method(print,eigcv)
+export("%>%")
 export(eigcv)
 import(Matrix)
-importFrom(dplyr,group_by)
-importFrom(dplyr,mutate)
-importFrom(dplyr,select)
-importFrom(dplyr,summarise)
-importFrom(dplyr,summarize)
-importFrom(dplyr,ungroup)
-importFrom(ggplot2,aes)
-importFrom(ggplot2,geom_hline)
-importFrom(ggplot2,geom_line)
-importFrom(ggplot2,geom_point)
-importFrom(ggplot2,ggplot)
-importFrom(ggplot2,labs)
-importFrom(ggplot2,scale_color_manual)
-importFrom(ggplot2,scale_x_continuous)
-importFrom(ggplot2,theme)
-importFrom(ggplot2,theme_bw)
 importFrom(magrittr,"%>%")
-importFrom(rlang,.data)
diff --git a/R/eigcv.R b/R/eigcv.R
index 55addb1..dae3660 100644
--- a/R/eigcv.R
+++ b/R/eigcv.R
@@ -1,9 +1,10 @@
+
+# assumes elements of A are non-negative integers (and Poisson)
 split_graph <- function(A, test_portion = 0.1) {
 
   A <- methods::as(A, "dMatrix")
   A <- methods::as(A, "TsparseMatrix")
 
-  # assumes elements of A are Poisson (i.e. non-negative integers)
   stopifnot(isTRUE(all.equal(A@x, as.integer(A@x))))
   test_edges <- stats::rbinom(length(A@x), size = A@x, prob = test_portion)
 
@@ -41,9 +42,11 @@ glaplacian <- function(A, regularize = TRUE) {
   L
 }
 
-# Given the trained left/right singular vectors, compute the test statistic for
+# given the training singular vectors, compute the test statistic for
 # graph dimension.
 gdstat <- function(full, test, u, v, test_portion) {
+
+  # standard error calculation is different for directed and undirected graphs
   if (isSymmetric(full)) {
     se <- sqrt(2 * test_portion * as.numeric(t(u^2) %*% full %*% v^2) -
       test_portion * sum(diag(full) * u^2 * v^2))
@@ -59,41 +62,56 @@ gdstat <- function(full, test, u, v, test_portion) {
 
 
 
-#' Edge Bootstrapping and Splitting
+#' Compute cross-validate eigenvalues
 #'
-#' Estimate the graph dimension via eigenvalue cross-validation (EigCV).
+#' Estimate graph dimension via eigenvalue cross-validation (EigCV).
 #' A graph has dimension `k` if the first `k` eigenvectors of its adjacency
 #' matrix are correlated with its population eigenspace, and the others are not.
 #' Edge bootstrapping sub-samples the edges of the graph (without replacement).
 #' Edge splitting separates the edges into a training part and a testing part.
 #'
-#' @param A The adjacency matrix of graph. Must be non-negative integer valued.
-#' @param k_max `integer(1)`, number of eigenvectors to compute.
-#' @param num_bootstraps `integer(1)`, number of graph bootstraps, default to 10.
-#'   Graph bootstrapping is to account for the randomness in graph splitting,
-#'   rather than obtaining any statistic (as a traditional num_bootstraps does).
-#'   Hence, a small number (e.g., 3~10) of bootstraps usually suffices.
-#'   If `num_bootstraps>1`, the test statistics will be averaged across bootstraps
+#' @param A The adjacency matrix of graph. Must be non-negative and
+#'  integer valued.
+#' @param k_max The maximum dimension of the graph to consider. This many
+#'   eigenvectors are computed. Should be a non-negative integer smallish
+#'   relative the dimensions of `A`.
+#' @param ... Ignored.
+#' @param num_bootstraps The number of times to bootstrap the graph. Since
+#'   cross-validated eigenvalues are based on a random graph split, they
+#'   are themselves random. By repeatedly computing cross-validated eigenvalues
+#'   for different sample splits, the idea is to smooth away some of the
+#'   randomness due to the graph splits. A small number of bootstraps
+#'   (3 to 10) usually suffices. Defaults to `10`. Test statistics (i.e.
+#'   z-scores for cv eigenvalues) are averaged across bootstraps
 #'   and the p-values will be calculated based on the averaged statistics.
-#' @param alpha Significance level of each test, defaults to `0.05`.
-#'   This is used to cut off the dimension estimation.
-#' @param ptol `numeric(1)`, the tolerance of minimal p-value.
-#' @param regularize TODO
-#' @param test_portion TODO
-#' @inheritParams stats::p.adjust
-#' @param laplacian `logical(1)`, use the normalized and regularized adjacency
-#'   matrix (i.e. L)
-#'   This option is experimental and should be used with caution.
-#' @return A `eigcv` object, which contains:
-#'   \item{estimated_dimension}{inferred graph dimension.}
-#'   \item{summary}{summary table of the tests.}
-#'   \item{num_bootstraps}{number of bootstraps performed.}
-#'   \item{test_portion}{graph splitting probability used.}
-#'   \item{alpha}{significance level of each test.}
+#' @param test_portion The portion of the graph to put into the test graph,
+#'   as opposed to the training graph. Defaults to `0.1`. Must be strictly
+#'   between zero and one.
+#' @param alpha Significance level for hypothesis tests. Each dimension
+#'   `1, ..., k_max` is tested when estimating graph dimension, and the
+#'   overall graph dimension is taken to be the smallest number of dimensions
+#'   such that all the tests reject.
+#' @param method Method to adjust p-values for multiple testing. Must be
+#'   one of `"none"`, `"holm"`, `"hochberg"`, `"hommel"`, `"bonferroni"`,
+#'   `"BH"`, `"BY"`, or `"fdr"`. Passed to [stats::p.adjust()]. Defaults to
+#'   `"none"`.
+#' @param laplacian Logical value indicating where to compute cross-validated
+#'   eigenvalues for the degree-normalize graph Laplacian rather than the
+#'   graph adjacency matrix. Experimental and should be used with caution.
+#'   Defaults to `FALSE`.
+#' @param regularize Only applicable when `laplacian == TRUE`, in which case
+#'   this parameter controls whether or not the degree-normalized graph
+#'   Laplacian is regularized. Defaults to `TRUE`.
+#'
+#' @return A `eigcv` object, which is a list with the following named
+#'   elements.
+#'
+#'   - `estimated_dimension`: inferred graph dimension.
+#'   - `summary`: summary table of the tests.
+#'   - `num_bootstraps`: number of bootstraps performed.
+#'   - `test_portion`: graph splitting probability used.
+#'   - `alpha`: significance level of each test.
 #'
-#' @importFrom dplyr summarize group_by ungroup mutate summarise
-#' @importFrom magrittr %>%
-#' @importFrom rlang .data
 #' @export
 #'
 #' @examples
@@ -116,16 +134,19 @@ gdstat <- function(full, test, u, v, test_portion) {
 #'
 #' A <- sample_sparse(model)
 #'
-#' eigcv_result <- eigcv(A, k_max = 10)
-#' eigcv_result
+#' eigs<- eigcv(A, k_max = 10)
+#' eigs
+#'
+#' plot(eigs, type = "z-score")    # default
+#' plot(eigs, type = "adjacency")
+#' plot(eigs, type = "laplacian")
+#'
 #'
 eigcv <- function(A, k_max,
                   ...,
                   num_bootstraps = 10, test_portion = 0.1,
                   alpha = 0.05,
-                  ptol = .Machine$double.eps,
-                  method = c("holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr",
-                             "none"),
+                  method = c("none", "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr"),
                   laplacian = FALSE,
                   regularize = TRUE) {
   n <- min(dim(A))
@@ -182,29 +203,23 @@ eigcv <- function(A, k_max,
     pb$tick()
   }
 
-  ## summarize across CV/num_bootstraps
-  if (num_bootstraps > 1) {
-    cv_means <- cv_stats %>%
-      group_by(.data$k) %>%
-      summarise(
-        cv_lambda_A = mean(.data$cv_lambda_A),
-        cv_lambda_L = mean(.data$cv_lambda_L),
-        z = mean(.data$z)
-      ) %>%
-      ungroup()
-  } else {
-    cv_means <- cv_stats
-  }
+  cv_means <- cv_stats %>%
+    dplyr::group_by(k) %>%
+    dplyr::summarise(
+      cv_lambda_A = mean(cv_lambda_A),
+      cv_lambda_L = mean(cv_lambda_L),
+      z = mean(z)
+    ) %>%
+    dplyr::ungroup()
+
+
   cv_means <- cv_means %>%
-    mutate(
-      pvals = stats::pnorm(.data$z, lower.tail = FALSE),
-      pvals = pmax(.data$pvals, ptol)
-    ) ## avoid exact 0
+    dplyr::mutate(
+      pvals = stats::pnorm(z, lower.tail = FALSE)
+    )
 
-  ## correct for multiplicity
   cv_means$padj <- stats::p.adjust(cv_means$pvals, method = method)
 
-  ## inference
   criteria <- cv_means$padj
   k_stop <- which(criteria > alpha)
   k_infer <- ifelse(length(k_stop), min(k_stop) - 1, k_max)
@@ -221,69 +236,80 @@ eigcv <- function(A, k_max,
 }
 
 
-#' Print `eigcv`
+#' Print cross-validated eigenvalues
 #'
-#' @method print eigcv
-#'
-#' @param x an `eigcv` object.
+#' @param x An `eigcv` object created by a call to [eigcv()].
 #' @param ... Ignored.
 #' @export
+#'
+#' @inherit eigcv examples
+#' @return `x`, but invisibly.
+#'
+#' @method print eigcv
 print.eigcv <- function(x,  ...) {
   cat("Estimated graph dimension:\t", x$estimated_dimension, fill = TRUE)
-    cat("\nNumber of bootstraps:\t\t", x$num_bootstraps, fill = TRUE)
-    cat("Edge splitting probabaility:\t", x$test_portion, fill = TRUE)
-    cat("Significance level:\t\t", x$alpha, fill = TRUE)
-    cat("\n ------------ Summary of Tests ------------\n")
-    print(data.frame(x$summary[, -c(2, 3)]), row.names = FALSE)
-    cat(fill = TRUE)
+  cat("\nNumber of bootstraps:\t\t", x$num_bootstraps, fill = TRUE)
+  cat("Edge splitting probabaility:\t", x$test_portion, fill = TRUE)
+  cat("Significance level:\t\t", x$alpha, fill = TRUE)
+  cat("\n ------------ Summary of Tests ------------\n")
+  print(data.frame(x$summary[, -c(2, 3)]), row.names = FALSE)
+  cat(fill = TRUE)
+  invisible(x)
 }
 
-#' Plot `eigcv`
+#' Plot cross-validated eigenvalues
 #'
-#' @method plot eigcv
+#' @param x An `eigcv` object created by a call to [eigcv()].
+#' @param type Specifies what to plot. Must be one of the following options:
+#'
+#'    - `"z-score"`, in which case the Z-statistic test scores are plotted
+#'      for each value of `k` (i.e. dimension of the eigenspace).
+#'    - `"adjacency"` in which case the cross-validated eigenvalues of the
+#'      adjacency matrix are plotted for each value of `k`.
+#'    - `"laplacian"` in which case the cross-validated eigenvalues of the
+#'      graph Laplacian matrix are plotted for each value of `k`.
 #'
-#' @param x an `eigcv` object.
-#' @param type either "z", "A", or "L" to specify the y-axis of the plot.
-#'   If "z", plot the test statistics (asymptotic z score) for each k.
-#'   If "A", plot x'Ax for each eigenvector x.
-#'   If "L", plot x'Lx for each eigenvector x.
-#' @param threshold `numeric(1)`, cut-off of p-value (in log10), default to 2.
-#' @param ... ignored.
-#' @return Plot an `eigcv` object.
-#' @importFrom ggplot2 ggplot aes labs theme_bw theme scale_color_manual
-#' @importFrom ggplot2 geom_hline geom_point geom_line scale_x_continuous
-#' @importFrom magrittr %>%
-#' @importFrom dplyr select
-#' @importFrom rlang .data
+#' @param threshold Only used when `type == "z-score"`. Adds a horizontal
+#'   line at the value of `threshold`, which should be a numeric of length
+#'   one. Defaults to `2`.
+#' @param ... Ignored.
+#' @return A `ggplot2` object.
 #' @export
-plot.eigcv <- function(x, type = c("z", "A", "L"), threshold = 2, ...) {
+#' @method plot eigcv
+#'
+#' @inherit eigcv examples
+plot.eigcv <- function(x, type = c("z-score", "adjacency", "laplacian"), threshold = 2, ...) {
   stopifnot("Threshold of statistics must be greater than 0." = threshold > 0)
 
+  type <- rlang::arg_match(type)
+
   type <- type[1]
-  if (type == "z") {
-    dat <- x$summary %>% select(.data$k, val = .data$z)
+  if (type == "z-score") {
+    dat <- dplyr::select(x$summary, k, val = z)
     ylab <- "z score"
   }
-  if (type == "A") {
-    dat <- x$summary %>% select(.data$k, val = .data$cv_lambda_A)
+  if (type == "adjacency") {
+    dat <- dplyr::select(x$summary, k, val = cv_lambda_A)
     ylab <- "cross validated x' A x"
   }
-  if (type == "L") {
-    dat <- x$summary %>% select(.data$k, val = .data$cv_lambda_L)
+  if (type == "laplacian") {
+    dat <- dplyr::select(x$summary, k, val = cv_lambda_L)
     ylab <- "cross validated x' L x"
   }
 
-  g <- ggplot(aes(.data$k, .data$val), data = dat) +
-    geom_point(alpha = .8) +
-    geom_line(color = "blue") +
-    theme_bw() +
-    scale_x_continuous(breaks = function(x) {
-      unique(floor(pretty(seq(0, (max(x) + 1) * 1.1))))
-    })
-
-  if (type == "z") {
+  g <- ggplot2::ggplot(ggplot2::aes(k, val), data = dat) +
+    ggplot2::geom_point(alpha = .8) +
+    ggplot2::geom_line(color = "blue") +
+    ggplot2::theme_bw() +
+    ggplot2::scale_x_continuous(
+      breaks = function(x) {
+        unique(floor(pretty(seq(0, (max(x) + 1) * 1.1))))
+      }
+    )
+
+  if (type == "z-score") {
     g <- g +
-      geom_hline(
+      ggplot2::geom_hline(
         yintercept = threshold, alpha = .8,
         linetype = 2, color = "grey60", show.legend = TRUE
       )
diff --git a/R/utils.R b/R/utils.R
new file mode 100644
index 0000000..2eba9f2
--- /dev/null
+++ b/R/utils.R
@@ -0,0 +1,24 @@
+#' Pipe operator
+#'
+#' See \code{magrittr::\link[magrittr:pipe]{\%>\%}} for details.
+#'
+#' @name %>%
+#' @rdname pipe
+#' @keywords internal
+#' @export
+#' @importFrom magrittr %>%
+#' @usage lhs \%>\% rhs
+#' @param lhs A value or the magrittr placeholder.
+#' @param rhs A function call using the magrittr semantics.
+#' @return The result of calling `rhs(lhs)`.
+NULL
+
+utils::globalVariables(
+  c(
+    "cv_lambda_A",
+    "cv_lambda_L",
+    "k",
+    "val",
+    "z"
+  )
+)
diff --git a/README.Rmd b/README.Rmd
index 95df027..8dfbb0b 100644
--- a/README.Rmd
+++ b/README.Rmd
@@ -21,7 +21,7 @@ knitr::opts_chunk$set(
 [![CRAN status](https://www.r-pkg.org/badges/version/gdim)](https://CRAN.R-project.org/package=gdim)
 <!-- badges: end -->
 
-`gdim` estimates graph dimension using cross-validated eigenvalues, via the graph-splitting technique developed in <http://arxiv.org/abs/2108.03336>. Theoretically, the method works by computing a special type of cross-validated eigenvalue which follows a simple central limit theorem. This allows users to perform hypothesis tests on the rank of the graph.
+`gdim` estimates graph dimension using cross-validated eigenvalues, via the graph-splitting technique developed in <https://arxiv.org/abs/2108.03336>. Theoretically, the method works by computing a special type of cross-validated eigenvalue which follows a simple central limit theorem. This allows users to perform hypothesis tests on the rank of the graph.
 
 ## Installation
 
@@ -77,4 +77,4 @@ plot(eigcv_result)
 
 ## Reference
 
-Chen, Fan, Sebastien Roch, Karl Rohe, and Shuqi Yu. “Estimating Graph Dimension with Cross-Validated Eigenvalues.” ArXiv:2108.03336 [Cs, Math, Stat], August 6, 2021. http://arxiv.org/abs/2108.03336.
+Chen, Fan, Sebastien Roch, Karl Rohe, and Shuqi Yu. “Estimating Graph Dimension with Cross-Validated Eigenvalues.” ArXiv:2108.03336 [Cs, Math, Stat], August 6, 2021. https://arxiv.org/abs/2108.03336.
diff --git a/README.md b/README.md
index 2c099f6..afccfc8 100644
--- a/README.md
+++ b/README.md
@@ -14,7 +14,7 @@ status](https://www.r-pkg.org/badges/version/gdim)](https://CRAN.R-project.org/p
 
 `gdim` estimates graph dimension using cross-validated eigenvalues, via
 the graph-splitting technique developed in
-<http://arxiv.org/abs/2108.03336>. Theoretically, the method works by
+<https://arxiv.org/abs/2108.03336>. Theoretically, the method works by
 computing a special type of cross-validated eigenvalue which follows a
 simple central limit theorem. This allows users to perform hypothesis
 tests on the rank of the graph.
@@ -80,16 +80,16 @@ eigcv_result
 #> 
 #>  ------------ Summary of Tests ------------
 #>   k          z        pvals         padj
-#>   1 60.1118360 2.220446e-16 2.220446e-15
-#>   2 12.4766591 2.220446e-16 2.220446e-15
-#>   3 12.0401480 2.220446e-16 2.220446e-15
-#>   4 10.7393316 2.220446e-16 2.220446e-15
-#>   5  8.9024334 2.220446e-16 2.220446e-15
-#>   6 -1.1480263 8.745212e-01 1.000000e+00
-#>   7 -1.6158410 9.469357e-01 1.000000e+00
-#>   8 -0.9936334 8.397993e-01 1.000000e+00
-#>   9 -0.8575451 8.044281e-01 1.000000e+00
-#>  10 -1.1543480 8.758212e-01 1.000000e+00
+#>   1 59.7488180 2.220446e-16 2.220446e-15
+#>   2 12.9094629 2.220446e-16 2.220446e-15
+#>   3 11.8600427 2.220446e-16 2.220446e-15
+#>   4 11.9412340 2.220446e-16 2.220446e-15
+#>   5  9.0252520 2.220446e-16 2.220446e-15
+#>   6 -0.8512008 8.026711e-01 1.000000e+00
+#>   7 -0.8182195 7.933841e-01 1.000000e+00
+#>   8 -0.9912649 8.392219e-01 1.000000e+00
+#>   9 -0.9005808 8.160944e-01 1.000000e+00
+#>  10 -1.1677953 8.785553e-01 1.000000e+00
 ```
 
 In this example, `eigcv()` suggests `k=5`.
@@ -108,4 +108,4 @@ plot(eigcv_result)
 
 Chen, Fan, Sebastien Roch, Karl Rohe, and Shuqi Yu. “Estimating Graph
 Dimension with Cross-Validated Eigenvalues.” ArXiv:2108.03336 \[Cs,
-Math, Stat\], August 6, 2021. <http://arxiv.org/abs/2108.03336>.
+Math, Stat\], August 6, 2021. <https://arxiv.org/abs/2108.03336>.
diff --git a/cran-comments.md b/cran-comments.md
index 858617d..4d11deb 100644
--- a/cran-comments.md
+++ b/cran-comments.md
@@ -1,5 +1,5 @@
 ## R CMD check results
 
-0 errors | 0 warnings | 1 note
+0 errors | 0 warnings | 1 notes
 
 * This is a new release.
diff --git a/man/eigcv.Rd b/man/eigcv.Rd
index b96ad23..179d10c 100644
--- a/man/eigcv.Rd
+++ b/man/eigcv.Rd
@@ -2,7 +2,7 @@
 % Please edit documentation in R/eigcv.R
 \name{eigcv}
 \alias{eigcv}
-\title{Edge Bootstrapping and Splitting}
+\title{Compute cross-validate eigenvalues}
 \usage{
 eigcv(
   A,
@@ -11,50 +11,66 @@ eigcv(
   num_bootstraps = 10,
   test_portion = 0.1,
   alpha = 0.05,
-  ptol = .Machine$double.eps,
-  method = c("holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr", "none"),
+  method = c("none", "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr"),
   laplacian = FALSE,
   regularize = TRUE
 )
 }
 \arguments{
-\item{A}{The adjacency matrix of graph. Must be non-negative integer valued.}
+\item{A}{The adjacency matrix of graph. Must be non-negative and
+integer valued.}
 
-\item{k_max}{\code{integer(1)}, number of eigenvectors to compute.}
+\item{k_max}{The maximum dimension of the graph to consider. This many
+eigenvectors are computed. Should be a non-negative integer smallish
+relative the dimensions of \code{A}.}
 
-\item{num_bootstraps}{\code{integer(1)}, number of graph bootstraps, default to 10.
-Graph bootstrapping is to account for the randomness in graph splitting,
-rather than obtaining any statistic (as a traditional num_bootstraps does).
-Hence, a small number (e.g., 3~10) of bootstraps usually suffices.
-If \code{num_bootstraps>1}, the test statistics will be averaged across bootstraps
-and the p-values will be calculated based on the averaged statistics.}
+\item{...}{Ignored.}
 
-\item{test_portion}{TODO}
+\item{num_bootstraps}{The number of times to bootstrap the graph. Since
+cross-validated eigenvalues are based on a random graph split, they
+are themselves random. By repeatedly computing cross-validated eigenvalues
+for different sample splits, the idea is to smooth away some of the
+randomness due to the graph splits. A small number of bootstraps
+(3 to 10) usually suffices. Defaults to \code{10}. Test statistics (i.e.
+z-scores for cv eigenvalues) are averaged across bootstraps
+and the p-values will be calculated based on the averaged statistics.}
 
-\item{alpha}{Significance level of each test, defaults to \code{0.05}.
-This is used to cut off the dimension estimation.}
+\item{test_portion}{The portion of the graph to put into the test graph,
+as opposed to the training graph. Defaults to \code{0.1}. Must be strictly
+between zero and one.}
 
-\item{ptol}{\code{numeric(1)}, the tolerance of minimal p-value.}
+\item{alpha}{Significance level for hypothesis tests. Each dimension
+\verb{1, ..., k_max} is tested when estimating graph dimension, and the
+overall graph dimension is taken to be the smallest number of dimensions
+such that all the tests reject.}
 
-\item{method}{correction method, a \code{\link{character}} string.
-    Can be abbreviated.}
+\item{method}{Method to adjust p-values for multiple testing. Must be
+one of \code{"none"}, \code{"holm"}, \code{"hochberg"}, \code{"hommel"}, \code{"bonferroni"},
+\code{"BH"}, \code{"BY"}, or \code{"fdr"}. Passed to \code{\link[stats:p.adjust]{stats::p.adjust()}}. Defaults to
+\code{"none"}.}
 
-\item{laplacian}{\code{logical(1)}, use the normalized and regularized adjacency
-matrix (i.e. L)
-This option is experimental and should be used with caution.}
+\item{laplacian}{Logical value indicating where to compute cross-validated
+eigenvalues for the degree-normalize graph Laplacian rather than the
+graph adjacency matrix. Experimental and should be used with caution.
+Defaults to \code{FALSE}.}
 
-\item{regularize}{TODO}
+\item{regularize}{Only applicable when \code{laplacian == TRUE}, in which case
+this parameter controls whether or not the degree-normalized graph
+Laplacian is regularized. Defaults to \code{TRUE}.}
 }
 \value{
-A \code{eigcv} object, which contains:
-\item{estimated_dimension}{inferred graph dimension.}
-\item{summary}{summary table of the tests.}
-\item{num_bootstraps}{number of bootstraps performed.}
-\item{test_portion}{graph splitting probability used.}
-\item{alpha}{significance level of each test.}
+A \code{eigcv} object, which is a list with the following named
+elements.
+\itemize{
+\item \code{estimated_dimension}: inferred graph dimension.
+\item \code{summary}: summary table of the tests.
+\item \code{num_bootstraps}: number of bootstraps performed.
+\item \code{test_portion}: graph splitting probability used.
+\item \code{alpha}: significance level of each test.
+}
 }
 \description{
-Estimate the graph dimension via eigenvalue cross-validation (EigCV).
+Estimate graph dimension via eigenvalue cross-validation (EigCV).
 A graph has dimension \code{k} if the first \code{k} eigenvectors of its adjacency
 matrix are correlated with its population eigenspace, and the others are not.
 Edge bootstrapping sub-samples the edges of the graph (without replacement).
@@ -80,7 +96,12 @@ model <- sbm(
 
 A <- sample_sparse(model)
 
-eigcv_result <- eigcv(A, k_max = 10)
-eigcv_result
+eigs<- eigcv(A, k_max = 10)
+eigs
+
+plot(eigs, type = "z-score")    # default
+plot(eigs, type = "adjacency")
+plot(eigs, type = "laplacian")
+
 
 }
diff --git a/man/figures/README-unnamed-chunk-3-1.png b/man/figures/README-unnamed-chunk-3-1.png
index ef24dad..d179cec 100644
Binary files a/man/figures/README-unnamed-chunk-3-1.png and b/man/figures/README-unnamed-chunk-3-1.png differ
diff --git a/man/gdim-package.Rd b/man/gdim-package.Rd
index 1080e90..e00e090 100644
--- a/man/gdim-package.Rd
+++ b/man/gdim-package.Rd
@@ -18,7 +18,7 @@ Useful links:
 
 }
 \author{
-\strong{Maintainer}: Alex Hayes \email{alexpghayes@gmail.com} (\href{https://orcid.org/0000-0002-4985-5160}{ORCID})
+\strong{Maintainer}: Alex Hayes \email{alexpghayes@gmail.com} (\href{https://orcid.org/0000-0002-4985-5160}{ORCID}) [copyright holder]
 
 Authors:
 \itemize{
diff --git a/man/pipe.Rd b/man/pipe.Rd
new file mode 100644
index 0000000..5fa90fe
--- /dev/null
+++ b/man/pipe.Rd
@@ -0,0 +1,20 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/utils.R
+\name{\%>\%}
+\alias{\%>\%}
+\title{Pipe operator}
+\usage{
+lhs \%>\% rhs
+}
+\arguments{
+\item{lhs}{A value or the magrittr placeholder.}
+
+\item{rhs}{A function call using the magrittr semantics.}
+}
+\value{
+The result of calling \code{rhs(lhs)}.
+}
+\description{
+See \code{magrittr::\link[magrittr:pipe]{\%>\%}} for details.
+}
+\keyword{internal}
diff --git a/man/plot.eigcv.Rd b/man/plot.eigcv.Rd
index f9987d9..3c96057 100644
--- a/man/plot.eigcv.Rd
+++ b/man/plot.eigcv.Rd
@@ -2,25 +2,61 @@
 % Please edit documentation in R/eigcv.R
 \name{plot.eigcv}
 \alias{plot.eigcv}
-\title{Plot \code{eigcv}}
+\title{Plot cross-validated eigenvalues}
 \usage{
-\method{plot}{eigcv}(x, type = c("z", "A", "L"), threshold = 2, ...)
+\method{plot}{eigcv}(x, type = c("z-score", "adjacency", "laplacian"), threshold = 2, ...)
 }
 \arguments{
-\item{x}{an \code{eigcv} object.}
+\item{x}{An \code{eigcv} object created by a call to \code{\link[=eigcv]{eigcv()}}.}
 
-\item{type}{either "z", "A", or "L" to specify the y-axis of the plot.
-If "z", plot the test statistics (asymptotic z score) for each k.
-If "A", plot x'Ax for each eigenvector x.
-If "L", plot x'Lx for each eigenvector x.}
+\item{type}{Specifies what to plot. Must be one of the following options:
+\itemize{
+\item \code{"z-score"}, in which case the Z-statistic test scores are plotted
+for each value of \code{k} (i.e. dimension of the eigenspace).
+\item \code{"adjacency"} in which case the cross-validated eigenvalues of the
+adjacency matrix are plotted for each value of \code{k}.
+\item \code{"laplacian"} in which case the cross-validated eigenvalues of the
+graph Laplacian matrix are plotted for each value of \code{k}.
+}}
 
-\item{threshold}{\code{numeric(1)}, cut-off of p-value (in log10), default to 2.}
+\item{threshold}{Only used when \code{type == "z-score"}. Adds a horizontal
+line at the value of \code{threshold}, which should be a numeric of length
+one. Defaults to \code{2}.}
 
-\item{...}{ignored.}
+\item{...}{Ignored.}
 }
 \value{
-Plot an \code{eigcv} object.
+A \code{ggplot2} object.
 }
 \description{
-Plot \code{eigcv}
+Plot cross-validated eigenvalues
+}
+\examples{
+
+library(fastRG)
+
+set.seed(27)
+
+B <- matrix(0.1, 5, 5)
+diag(B) <- 0.3
+
+model <- sbm(
+  n = 1000,
+  k = 5,
+  B = B,
+  expected_degree = 40,
+  poisson_edges = FALSE,
+  allow_self_loops = FALSE
+)
+
+A <- sample_sparse(model)
+
+eigs<- eigcv(A, k_max = 10)
+eigs
+
+plot(eigs, type = "z-score")    # default
+plot(eigs, type = "adjacency")
+plot(eigs, type = "laplacian")
+
+
 }
diff --git a/man/print.eigcv.Rd b/man/print.eigcv.Rd
index 2a47a1d..e30249b 100644
--- a/man/print.eigcv.Rd
+++ b/man/print.eigcv.Rd
@@ -2,15 +2,47 @@
 % Please edit documentation in R/eigcv.R
 \name{print.eigcv}
 \alias{print.eigcv}
-\title{Print \code{eigcv}}
+\title{Print cross-validated eigenvalues}
 \usage{
 \method{print}{eigcv}(x, ...)
 }
 \arguments{
-\item{x}{an \code{eigcv} object.}
+\item{x}{An \code{eigcv} object created by a call to \code{\link[=eigcv]{eigcv()}}.}
 
 \item{...}{Ignored.}
 }
+\value{
+\code{x}, but invisibly.
+}
 \description{
-Print \code{eigcv}
+Print cross-validated eigenvalues
+}
+\examples{
+
+library(fastRG)
+
+set.seed(27)
+
+B <- matrix(0.1, 5, 5)
+diag(B) <- 0.3
+
+model <- sbm(
+  n = 1000,
+  k = 5,
+  B = B,
+  expected_degree = 40,
+  poisson_edges = FALSE,
+  allow_self_loops = FALSE
+)
+
+A <- sample_sparse(model)
+
+eigs<- eigcv(A, k_max = 10)
+eigs
+
+plot(eigs, type = "z-score")    # default
+plot(eigs, type = "adjacency")
+plot(eigs, type = "laplacian")
+
+
 }