diff --git a/NAMESPACE b/NAMESPACE
index 95be07fb203..2b491c6d8d1 100644
--- a/NAMESPACE
+++ b/NAMESPACE
@@ -160,6 +160,7 @@ export(as_phylo)
 export(as_star)
 export(as_tree)
 export(as_undirected)
+export(as_veincidence_matrix)
 export(assortativity)
 export(assortativity.degree)
 export(assortativity.nominal)
diff --git a/R/conversion.R b/R/conversion.R
index 4de5467dd4c..7f19367ff62 100644
--- a/R/conversion.R
+++ b/R/conversion.R
@@ -498,6 +498,297 @@ as_edgelist <- function(graph, names = TRUE) {
   res
 }
 
+#' Convert a graph to a vertex-edge incidence matrix
+#'
+#' `r lifecycle::badge("experimental")`
+#'
+#' `as_veincidence_matrix()` returns the vertex-edge incidence matrix of a graph.
+#'
+#' The vertex-edge incidence matrix is a matrix where rows correspond to vertices
+#' and columns correspond to edges. For undirected graphs, the matrix element
+#' is 1 if the vertex is incident to the edge, 0 otherwise. For directed graphs,
+#' the element is -1 if the vertex is the tail (source) of the edge, +1 if it is
+#' the head (target), and 0 otherwise.
+#'
+#' For loops (edges connecting a vertex to itself):
+#' \itemize{
+#'   \item In undirected graphs: the incidence value is 2
+#'   \item In directed graphs: the incidence value is 0
+#' }
+#'
+#' When weights are used, the values -1 and +1 are replaced by -w and +w
+#' respectively, where w is the weight of the edge.
+#'
+#' @param graph The input graph.
+#' @param types `r lifecycle::badge("deprecated")` This argument is deprecated
+#'   and not used. It was originally intended for compatibility but vertex-edge
+#'   incidence matrices don't use types.
+#' @param attr Either `NULL` or a character string giving an edge attribute name.
+#'   If `NULL`, unweighted incidence is returned (values are -1, 0, 1 for directed
+#'   graphs and 0, 1, 2 for undirected graphs). If not `NULL`, weighted incidence
+#'   is returned using the specified edge attribute as weights. For directed graphs,
+#'   values are -w, 0, w where w is the weight. For undirected graphs, values are
+#'   0, w, 2w.
+#' @param names Logical scalar, whether to add vertex and edge names to the matrix.
+#'   If `TRUE` (default), vertex names are used for row names if the `name`
+#'   vertex attribute exists, and edge labels are used for column names if
+#'   the `label` edge attribute exists. Vertex and edge IDs are used otherwise.
+#' @param sparse Logical scalar, whether to return a sparse matrix. The
+#'   \sQuote{`Matrix`} package must be installed for creating sparse matrices.
+#' @return A (usually sparse) matrix with `vcount(graph)` rows and
+#'   `ecount(graph)` columns.
+#'
+#' @seealso [as_adjacency_matrix()], [as_edgelist()], [laplacian_matrix()]
+#'
+#' @examples
+#' # Undirected graph
+#' g1 <- make_ring(3, circular = FALSE)
+#' as_veincidence_matrix(g1, sparse = FALSE)
+#'
+#' # Directed graph
+#' g2 <- make_ring(3, circular = FALSE, directed = TRUE)
+#' as_veincidence_matrix(g2, sparse = FALSE)
+#'
+#' # Weighted graph
+#' g3 <- make_ring(3, circular = FALSE)
+#' E(g3)$weight <- c(2, 3)
+#' as_veincidence_matrix(g3, attr = "weight", sparse = FALSE)
+#'
+#' # Named graph
+#' g4 <- graph_from_literal(a-b, b-c, c-d)
+#' E(g4)$label <- paste0("e", seq_len(ecount(g4)))
+#' as_veincidence_matrix(g4, sparse = FALSE)
+#'
+#' # Graph with loop
+#' g5 <- graph(c(1, 1, 1, 2), directed = FALSE)
+#' as_veincidence_matrix(g5, sparse = FALSE)
+#'
+#' @family conversion
+#' @concept matrices
+#' @export
+as_veincidence_matrix <- function(
+  graph,
+  types = deprecated(),
+  attr = NULL,
+  names = TRUE,
+  sparse = igraph_opt("sparsematrices")
+) {
+  ensure_igraph(graph)
+
+  if (lifecycle::is_present(types)) {
+    lifecycle::deprecate_warn(
+      "2.1.0",
+      "as_veincidence_matrix(types = )",
+      details = "The 'types' argument is not used for vertex-edge incidence matrices."
+    )
+  }
+
+  if (sparse) {
+    get.ve.incidence.sparse(graph, attr = attr, names = names)
+  } else {
+    get.ve.incidence.dense(graph, attr = attr, names = names)
+  }
+}
+
+get.ve.incidence.dense <- function(
+  graph,
+  attr = NULL,
+  names = TRUE,
+  call = rlang::caller_env()
+) {
+  vc <- vcount(graph)
+  ec <- ecount(graph)
+
+  # Initialize matrix with zeros
+  res <- matrix(0, nrow = vc, ncol = ec)
+
+  if (ec == 0) {
+    # Empty graph
+    if (names && "name" %in% vertex_attr_names(graph)) {
+      rownames(res) <- V(graph)$name
+    }
+    return(res)
+  }
+
+  # Get edge list
+  el <- as_edgelist(graph, names = FALSE)
+
+  # Get weights if specified
+  if (!is.null(attr)) {
+    attr <- as.character(attr)
+    if (!attr %in% edge_attr_names(graph)) {
+      cli::cli_abort("No such edge attribute: {.str {attr}}", call = call)
+    }
+    weights <- edge_attr(graph, attr)
+    if (!is.numeric(weights)) {
+      cli::cli_abort(
+        "Edge attribute {.str {attr}} must be numeric",
+        call = call
+      )
+    }
+  } else {
+    weights <- rep(1, ec)
+  }
+
+  if (is_directed(graph)) {
+    # For directed graphs: -weight for tail, +weight for head
+    for (i in seq_len(ec)) {
+      tail <- el[i, 1]
+      head <- el[i, 2]
+
+      if (tail == head) {
+        # Self-loop: contribution is 0 in directed graphs
+        res[tail, i] <- 0
+      } else {
+        res[tail, i] <- -weights[i]
+        res[head, i] <- weights[i]
+      }
+    }
+  } else {
+    # For undirected graphs: weight for each incident vertex
+    for (i in seq_len(ec)) {
+      v1 <- el[i, 1]
+      v2 <- el[i, 2]
+
+      if (v1 == v2) {
+        # Self-loop: contribution is 2*weight
+        res[v1, i] <- 2 * weights[i]
+      } else {
+        res[v1, i] <- weights[i]
+        res[v2, i] <- weights[i]
+      }
+    }
+  }
+
+  # Add row names (vertex names or IDs)
+  if (names) {
+    if ("name" %in% vertex_attr_names(graph)) {
+      rownames(res) <- V(graph)$name
+    } else {
+      rownames(res) <- seq_len(vc)
+    }
+
+    # Add column names (edge labels or IDs)
+    if (
+      !is.null(attr) && attr == "label" && "label" %in% edge_attr_names(graph)
+    ) {
+      colnames(res) <- E(graph)$label
+    } else if ("label" %in% edge_attr_names(graph)) {
+      colnames(res) <- E(graph)$label
+    } else {
+      colnames(res) <- seq_len(ec)
+    }
+  }
+
+  res
+}
+
+get.ve.incidence.sparse <- function(
+  graph,
+  attr = NULL,
+  names = TRUE,
+  call = rlang::caller_env()
+) {
+  vc <- vcount(graph)
+  ec <- ecount(graph)
+
+  if (ec == 0) {
+    # Empty graph
+    res <- Matrix::Matrix(0, nrow = vc, ncol = ec, sparse = TRUE)
+    if (names && "name" %in% vertex_attr_names(graph)) {
+      rownames(res) <- V(graph)$name
+    }
+    return(res)
+  }
+
+  # Get edge list
+  el <- as_edgelist(graph, names = FALSE)
+
+  # Get weights if specified
+  if (!is.null(attr)) {
+    attr <- as.character(attr)
+    if (!attr %in% edge_attr_names(graph)) {
+      cli::cli_abort("No such edge attribute: {.str {attr}}", call = call)
+    }
+    weights <- edge_attr(graph, attr)
+    if (!is.numeric(weights)) {
+      cli::cli_abort(
+        "Edge attribute {.str {attr}} must be numeric",
+        call = call
+      )
+    }
+  } else {
+    weights <- rep(1, ec)
+  }
+
+  # Build sparse matrix using triplet format
+  i_indices <- integer()
+  j_indices <- integer()
+  x_values <- numeric()
+
+  if (is_directed(graph)) {
+    # For directed graphs: -weight for tail, +weight for head
+    for (e in seq_len(ec)) {
+      tail <- el[e, 1]
+      head <- el[e, 2]
+
+      if (tail == head) {
+        # Self-loop: contribution is 0, so we skip it (sparse)
+      } else {
+        i_indices <- c(i_indices, tail, head)
+        j_indices <- c(j_indices, e, e)
+        x_values <- c(x_values, -weights[e], weights[e])
+      }
+    }
+  } else {
+    # For undirected graphs: weight for each incident vertex
+    for (e in seq_len(ec)) {
+      v1 <- el[e, 1]
+      v2 <- el[e, 2]
+
+      if (v1 == v2) {
+        # Self-loop: contribution is 2*weight
+        i_indices <- c(i_indices, v1)
+        j_indices <- c(j_indices, e)
+        x_values <- c(x_values, 2 * weights[e])
+      } else {
+        i_indices <- c(i_indices, v1, v2)
+        j_indices <- c(j_indices, e, e)
+        x_values <- c(x_values, weights[e], weights[e])
+      }
+    }
+  }
+
+  # Create sparse matrix
+  res <- Matrix::sparseMatrix(
+    i = i_indices,
+    j = j_indices,
+    x = x_values,
+    dims = c(vc, ec)
+  )
+
+  # Add row names (vertex names or IDs)
+  if (names) {
+    if ("name" %in% vertex_attr_names(graph)) {
+      rownames(res) <- V(graph)$name
+    } else {
+      rownames(res) <- seq_len(vc)
+    }
+
+    # Add column names (edge labels or IDs)
+    if (
+      !is.null(attr) && attr == "label" && "label" %in% edge_attr_names(graph)
+    ) {
+      colnames(res) <- E(graph)$label
+    } else if ("label" %in% edge_attr_names(graph)) {
+      colnames(res) <- E(graph)$label
+    } else {
+      colnames(res) <- seq_len(ec)
+    }
+  }
+
+  res
+}
 
 #' Convert between directed and undirected graphs
 #'
diff --git a/man/as.matrix.igraph.Rd b/man/as.matrix.igraph.Rd
index 136377aa4c0..0d2eba60b50 100644
--- a/man/as.matrix.igraph.Rd
+++ b/man/as.matrix.igraph.Rd
@@ -55,6 +55,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_adj_list.Rd b/man/as_adj_list.Rd
index 24d17623734..e0a0d617f2f 100644
--- a/man/as_adj_list.Rd
+++ b/man/as_adj_list.Rd
@@ -74,6 +74,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_adjacency_matrix.Rd b/man/as_adjacency_matrix.Rd
index 5f6215049b0..d288c3230d1 100644
--- a/man/as_adjacency_matrix.Rd
+++ b/man/as_adjacency_matrix.Rd
@@ -81,6 +81,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_biadjacency_matrix.Rd b/man/as_biadjacency_matrix.Rd
index c4cd1cd8ca3..9a77a8d2f7f 100644
--- a/man/as_biadjacency_matrix.Rd
+++ b/man/as_biadjacency_matrix.Rd
@@ -70,6 +70,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_directed.Rd b/man/as_directed.Rd
index 78e4fdfe586..9ef4861791d 100644
--- a/man/as_directed.Rd
+++ b/man/as_directed.Rd
@@ -129,6 +129,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_edgelist.Rd b/man/as_edgelist.Rd
index 9708cb79784..b1de4a2b1b1 100644
--- a/man/as_edgelist.Rd
+++ b/man/as_edgelist.Rd
@@ -44,6 +44,7 @@ Other conversion:
 \code{\link{as_directed}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_graphnel.Rd b/man/as_graphnel.Rd
index 96650d6746f..b547b34da5b 100644
--- a/man/as_graphnel.Rd
+++ b/man/as_graphnel.Rd
@@ -56,6 +56,7 @@ Other conversion:
 \code{\link{as_directed}()},
 \code{\link{as_edgelist}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_long_data_frame.Rd b/man/as_long_data_frame.Rd
index 4e5e865dee9..b86128ffe2f 100644
--- a/man/as_long_data_frame.Rd
+++ b/man/as_long_data_frame.Rd
@@ -40,6 +40,7 @@ Other conversion:
 \code{\link{as_directed}()},
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 }
diff --git a/man/as_veincidence_matrix.Rd b/man/as_veincidence_matrix.Rd
new file mode 100644
index 00000000000..c652fe9c9c0
--- /dev/null
+++ b/man/as_veincidence_matrix.Rd
@@ -0,0 +1,103 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/conversion.R
+\name{as_veincidence_matrix}
+\alias{as_veincidence_matrix}
+\title{Convert a graph to a vertex-edge incidence matrix}
+\usage{
+as_veincidence_matrix(
+  graph,
+  types = deprecated(),
+  attr = NULL,
+  names = TRUE,
+  sparse = igraph_opt("sparsematrices")
+)
+}
+\arguments{
+\item{graph}{The input graph.}
+
+\item{types}{\ifelse{html}{\href{https://lifecycle.r-lib.org/articles/stages.html#deprecated}{\figure{lifecycle-deprecated.svg}{options: alt='[Deprecated]'}}}{\strong{[Deprecated]}} This argument is deprecated
+and not used. It was originally intended for compatibility but vertex-edge
+incidence matrices don't use types.}
+
+\item{attr}{Either \code{NULL} or a character string giving an edge attribute name.
+If \code{NULL}, unweighted incidence is returned (values are -1, 0, 1 for directed
+graphs and 0, 1, 2 for undirected graphs). If not \code{NULL}, weighted incidence
+is returned using the specified edge attribute as weights. For directed graphs,
+values are -w, 0, w where w is the weight. For undirected graphs, values are
+0, w, 2w.}
+
+\item{names}{Logical scalar, whether to add vertex and edge names to the matrix.
+If \code{TRUE} (default), vertex names are used for row names if the \code{name}
+vertex attribute exists, and edge labels are used for column names if
+the \code{label} edge attribute exists. Vertex and edge IDs are used otherwise.}
+
+\item{sparse}{Logical scalar, whether to return a sparse matrix. The
+\sQuote{\code{Matrix}} package must be installed for creating sparse matrices.}
+}
+\value{
+A (usually sparse) matrix with \code{vcount(graph)} rows and
+\code{ecount(graph)} columns.
+}
+\description{
+\ifelse{html}{\href{https://lifecycle.r-lib.org/articles/stages.html#experimental}{\figure{lifecycle-experimental.svg}{options: alt='[Experimental]'}}}{\strong{[Experimental]}}
+}
+\details{
+\code{as_veincidence_matrix()} returns the vertex-edge incidence matrix of a graph.
+
+The vertex-edge incidence matrix is a matrix where rows correspond to vertices
+and columns correspond to edges. For undirected graphs, the matrix element
+is 1 if the vertex is incident to the edge, 0 otherwise. For directed graphs,
+the element is -1 if the vertex is the tail (source) of the edge, +1 if it is
+the head (target), and 0 otherwise.
+
+For loops (edges connecting a vertex to itself):
+\itemize{
+\item In undirected graphs: the incidence value is 2
+\item In directed graphs: the incidence value is 0
+}
+
+When weights are used, the values -1 and +1 are replaced by -w and +w
+respectively, where w is the weight of the edge.
+}
+\examples{
+# Undirected graph
+g1 <- make_ring(3, circular = FALSE)
+as_veincidence_matrix(g1, sparse = FALSE)
+
+# Directed graph
+g2 <- make_ring(3, circular = FALSE, directed = TRUE)
+as_veincidence_matrix(g2, sparse = FALSE)
+
+# Weighted graph
+g3 <- make_ring(3, circular = FALSE)
+E(g3)$weight <- c(2, 3)
+as_veincidence_matrix(g3, attr = "weight", sparse = FALSE)
+
+# Named graph
+g4 <- graph_from_literal(a-b, b-c, c-d)
+E(g4)$label <- paste0("e", seq_len(ecount(g4)))
+as_veincidence_matrix(g4, sparse = FALSE)
+
+# Graph with loop
+g5 <- graph(c(1, 1, 1, 2), directed = FALSE)
+as_veincidence_matrix(g5, sparse = FALSE)
+
+}
+\seealso{
+\code{\link[=as_adjacency_matrix]{as_adjacency_matrix()}}, \code{\link[=as_edgelist]{as_edgelist()}}, \code{\link[=laplacian_matrix]{laplacian_matrix()}}
+
+Other conversion: 
+\code{\link{as.matrix.igraph}()},
+\code{\link{as_adj_list}()},
+\code{\link{as_adjacency_matrix}()},
+\code{\link{as_biadjacency_matrix}()},
+\code{\link{as_data_frame}()},
+\code{\link{as_directed}()},
+\code{\link{as_edgelist}()},
+\code{\link{as_graphnel}()},
+\code{\link{as_long_data_frame}()},
+\code{\link{graph_from_adj_list}()},
+\code{\link{graph_from_graphnel}()}
+}
+\concept{conversion}
+\concept{matrices}
diff --git a/man/graph_from_adj_list.Rd b/man/graph_from_adj_list.Rd
index f3881bdf42a..fd995444016 100644
--- a/man/graph_from_adj_list.Rd
+++ b/man/graph_from_adj_list.Rd
@@ -75,6 +75,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_graphnel}()}
 }
 \author{
diff --git a/man/graph_from_data_frame.Rd b/man/graph_from_data_frame.Rd
index 6aa7caeddbb..226264c43b7 100644
--- a/man/graph_from_data_frame.Rd
+++ b/man/graph_from_data_frame.Rd
@@ -137,6 +137,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()},
 \code{\link{graph_from_graphnel}()}
 
diff --git a/man/graph_from_graphnel.Rd b/man/graph_from_graphnel.Rd
index ef7b6d0f1ed..afed3ad6470 100644
--- a/man/graph_from_graphnel.Rd
+++ b/man/graph_from_graphnel.Rd
@@ -71,6 +71,7 @@ Other conversion:
 \code{\link{as_edgelist}()},
 \code{\link{as_graphnel}()},
 \code{\link{as_long_data_frame}()},
+\code{\link{as_veincidence_matrix}()},
 \code{\link{graph_from_adj_list}()}
 }
 \concept{conversion}
diff --git a/tests/testthat/test-ve-incidence.R b/tests/testthat/test-ve-incidence.R
new file mode 100644
index 00000000000..ff7bfd9bc38
--- /dev/null
+++ b/tests/testthat/test-ve-incidence.R
@@ -0,0 +1,269 @@
+test_that("as_veincidence_matrix works for simple undirected graph", {
+  g <- make_ring(3, circular = FALSE)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 3)
+  expect_equal(ncol(inc), 2)
+  
+  # Vertices 1 and 2 are incident to edge 1
+  expect_equal(inc[1, 1], 1)
+  expect_equal(inc[2, 1], 1)
+  expect_equal(inc[3, 1], 0)
+  
+  # Vertices 2 and 3 are incident to edge 2
+  expect_equal(inc[1, 2], 0)
+  expect_equal(inc[2, 2], 1)
+  expect_equal(inc[3, 2], 1)
+})
+
+test_that("as_veincidence_matrix works for simple directed graph", {
+  g <- make_ring(3, circular = FALSE, directed = TRUE)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 3)
+  expect_equal(ncol(inc), 2)
+  
+  # Edge 1 goes from vertex 1 to vertex 2
+  expect_equal(inc[1, 1], -1)
+  expect_equal(inc[2, 1], 1)
+  expect_equal(inc[3, 1], 0)
+  
+  # Edge 2 goes from vertex 2 to vertex 3
+  expect_equal(inc[1, 2], 0)
+  expect_equal(inc[2, 2], -1)
+  expect_equal(inc[3, 2], 1)
+})
+
+test_that("as_veincidence_matrix works with weights for undirected graph", {
+  g <- make_ring(3, circular = FALSE)
+  E(g)$weight <- c(2, 3)
+  inc <- as_veincidence_matrix(g, attr = "weight", sparse = FALSE)
+  
+  expect_equal(inc[1, 1], 2)
+  expect_equal(inc[2, 1], 2)
+  expect_equal(inc[2, 2], 3)
+  expect_equal(inc[3, 2], 3)
+})
+
+test_that("as_veincidence_matrix works with weights for directed graph", {
+  g <- make_ring(3, circular = FALSE, directed = TRUE)
+  E(g)$weight <- c(2, 5)
+  inc <- as_veincidence_matrix(g, attr = "weight", sparse = FALSE)
+  
+  expect_equal(inc[1, 1], -2)
+  expect_equal(inc[2, 1], 2)
+  expect_equal(inc[2, 2], -5)
+  expect_equal(inc[3, 2], 5)
+})
+
+test_that("as_veincidence_matrix works with vertex names", {
+  g <- graph_from_literal(a-b, b-c, c-d)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(rownames(inc), c("a", "b", "c", "d"))
+  expect_equal(ncol(inc), 3)
+})
+
+test_that("as_veincidence_matrix works with edge labels", {
+  g <- graph_from_literal(a-b, b-c)
+  E(g)$label <- c("e1", "e2")
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(colnames(inc), c("e1", "e2"))
+})
+
+test_that("as_veincidence_matrix works with undirected loops", {
+  g <- make_graph(c(1, 1, 1, 2), directed = FALSE)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 2)
+  expect_equal(ncol(inc), 2)
+  
+  # First edge is a loop on vertex 1, should have value 2
+  expect_equal(inc[1, 1], 2)
+  expect_equal(inc[2, 1], 0)
+  
+  # Second edge connects vertices 1 and 2
+  expect_equal(inc[1, 2], 1)
+  expect_equal(inc[2, 2], 1)
+})
+
+test_that("as_veincidence_matrix works with directed loops", {
+  g <- make_graph(c(1, 1, 1, 2), directed = TRUE)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 2)
+  expect_equal(ncol(inc), 2)
+  
+  # First edge is a loop on vertex 1, should have value 0
+  expect_equal(inc[1, 1], 0)
+  expect_equal(inc[2, 1], 0)
+  
+  # Second edge goes from vertex 1 to vertex 2
+  expect_equal(inc[1, 2], -1)
+  expect_equal(inc[2, 2], 1)
+})
+
+test_that("as_veincidence_matrix works with sparse matrices", {
+  g <- make_ring(5, circular = FALSE)
+  inc_sparse <- as_veincidence_matrix(g, sparse = TRUE)
+  inc_dense <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_true(inherits(inc_sparse, "Matrix"))
+  expect_true(inherits(inc_sparse, "sparseMatrix"))
+  expect_equal(as.matrix(inc_sparse), inc_dense)
+})
+
+test_that("as_veincidence_matrix works with multi-edges", {
+  g <- make_graph(c(1, 2, 1, 2, 2, 3), directed = FALSE)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 3)
+  expect_equal(ncol(inc), 3)
+  
+  # Two edges between vertices 1 and 2
+  expect_equal(inc[1, 1], 1)
+  expect_equal(inc[2, 1], 1)
+  expect_equal(inc[1, 2], 1)
+  expect_equal(inc[2, 2], 1)
+  
+  # One edge between vertices 2 and 3
+  expect_equal(inc[2, 3], 1)
+  expect_equal(inc[3, 3], 1)
+})
+
+test_that("as_veincidence_matrix works with weighted multi-edges", {
+  g <- make_graph(c(1, 2, 1, 2, 2, 3), directed = TRUE)
+  E(g)$weight <- c(2, 3, 5)
+  inc <- as_veincidence_matrix(g, attr = "weight", sparse = FALSE)
+  
+  expect_equal(inc[1, 1], -2)
+  expect_equal(inc[2, 1], 2)
+  expect_equal(inc[1, 2], -3)
+  expect_equal(inc[2, 2], 3)
+  expect_equal(inc[2, 3], -5)
+  expect_equal(inc[3, 3], 5)
+})
+
+test_that("as_veincidence_matrix works with empty graph", {
+  g <- make_empty_graph(5)
+  inc <- as_veincidence_matrix(g, sparse = FALSE)
+  
+  expect_equal(nrow(inc), 5)
+  expect_equal(ncol(inc), 0)
+})
+
+test_that("as_veincidence_matrix fails with non-numeric edge attribute", {
+  g <- make_ring(3, circular = FALSE)
+  E(g)$label <- c("a", "b")
+  
+  expect_error(
+    as_veincidence_matrix(g, attr = "label"),
+    "must be numeric"
+  )
+})
+
+test_that("as_veincidence_matrix fails with non-existent edge attribute", {
+  g <- make_ring(3, circular = FALSE)
+  
+  expect_error(
+    as_veincidence_matrix(g, attr = "nonexistent"),
+    "No such edge attribute"
+  )
+})
+
+test_that("as_veincidence_matrix relationship with Laplacian", {
+  # For a graph's incidence matrix B:
+  # - For undirected graphs: B * B^T = D + A (signless Laplacian), not D - A
+  # - For directed graphs: B * B^T gives a symmetric matrix (not the out-Laplacian)
+  g <- make_ring(4, circular = TRUE)
+  
+  B <- as_veincidence_matrix(g, sparse = FALSE, names = FALSE)
+  
+  # For undirected graphs: B * B^T gives the signless Laplacian Q = D + A
+  Q_computed <- B %*% t(B)
+  
+  # Verify it's positive semidefinite (all diagonal elements are positive)
+  expect_true(all(diag(Q_computed) > 0))
+  
+  # Verify dimensions
+  expect_equal(nrow(B), vcount(g))
+  expect_equal(ncol(B), ecount(g))
+  expect_equal(nrow(Q_computed), vcount(g))
+  expect_equal(ncol(Q_computed), vcount(g))
+  
+  # Verify symmetry
+  expect_true(isSymmetric(Q_computed))
+})
+
+test_that("as_veincidence_matrix relationship with weighted Laplacian", {
+  # For weighted graphs, just verify dimensions and basic properties
+  g <- make_ring(4, circular = TRUE)
+  E(g)$weight <- c(1, 2, 3, 4)
+  
+  B_w <- as_veincidence_matrix(g, attr = "weight", sparse = FALSE)
+  L <- laplacian_matrix(g, weights = E(g)$weight, sparse = FALSE)
+  
+  # Verify dimensions match
+  expect_equal(nrow(B_w), vcount(g))
+  expect_equal(ncol(B_w), ecount(g))
+  expect_equal(nrow(L), vcount(g))
+  expect_equal(ncol(L), vcount(g))
+  
+  # Verify B_w * B_w^T is symmetric for undirected graphs
+  Q <- B_w %*% t(B_w)
+  expect_true(isSymmetric(Q))
+})
+
+test_that("as_veincidence_matrix names parameter works", {
+  g <- graph_from_literal(a-b, b-c)
+  E(g)$label <- c("e1", "e2")
+  
+  inc_named <- as_veincidence_matrix(g, names = TRUE, sparse = FALSE)
+  inc_unnamed <- as_veincidence_matrix(g, names = FALSE, sparse = FALSE)
+  
+  expect_equal(rownames(inc_named), c("a", "b", "c"))
+  expect_equal(colnames(inc_named), c("e1", "e2"))
+  expect_null(rownames(inc_unnamed))
+  expect_null(colnames(inc_unnamed))
+})
+
+test_that("as_veincidence_matrix examples from issue work correctly", {
+  # Example g2: Undirected, multi edge
+  g2 <- make_graph(c(1, 2, 1, 2, 2, 3), directed = FALSE)
+  inc2 <- as_veincidence_matrix(g2, sparse = FALSE, names = FALSE)
+  
+  expected2 <- matrix(c(1, 1, 0, 1, 1, 0, 0, 1, 1), nrow = 3, ncol = 3)
+  expect_equal(inc2, expected2)
+  
+  # Example g4w: Directed, multi edge, weighted
+  g4w <- make_graph(c(1, 2, 1, 2, 2, 3), directed = TRUE)
+  E(g4w)$weight <- c(2, 3, 5)
+  E(g4w)$label <- c("2", "3", "5")
+  inc4w <- as_veincidence_matrix(g4w, attr = "weight", sparse = FALSE, names = FALSE)
+  
+  expected4w <- matrix(c(-2, 2, 0, -3, 3, 0, 0, -5, 5), nrow = 3, ncol = 3)
+  expect_equal(inc4w, expected4w)
+  
+  # Example g6: Named graph, directed
+  g6 <- graph_from_literal(a-+b-+c-+a, a-+d, simplify = FALSE)
+  E(g6)$label <- c("1", "2", "3", "4")
+  inc6 <- as_veincidence_matrix(g6, sparse = FALSE)
+  
+  expect_equal(rownames(inc6), c("a", "b", "c", "d"))
+  expect_equal(colnames(inc6), c("1", "2", "3", "4"))
+  
+  # Example g7: Undirected graph with loops
+  g7 <- make_graph(c(1, 1, 1, 1, 1, 2, 1, 2), directed = FALSE)
+  inc7 <- as_veincidence_matrix(g7, sparse = FALSE, names = FALSE)
+  
+  expected7 <- matrix(c(2, 0, 2, 0, 1, 1, 1, 1), nrow = 2, ncol = 4)
+  expect_equal(inc7, expected7)
+  
+  # Example g8: Directed graph with loops
+  g8 <- make_graph(c(1, 1, 1, 1, 1, 2, 1, 2), directed = TRUE)
+  inc8 <- as_veincidence_matrix(g8, sparse = FALSE, names = FALSE)
+  
+  expected8 <- matrix(c(0, 0, 0, 0, -1, 1, -1, 1), nrow = 2, ncol = 4)
+  expect_equal(inc8, expected8)
+})