directed – Mirror image of graph

I need to display a directed acyclic graph (say Hasse diagram of partial order) oriented with minimal elements at the bottom, maximal elements at the top as is conventional for Hasse diagrams.

I tried it for:

graph1 = Graph({b (DirectedEdge) a, c (DirectedEdge) a, d -> b, 

e -> b}, EdgeStyle -> Arrowheads(.04),
VertexLabels -> Placed(“Name”, Center), VertexSize -> 0.2,
VertexStyle -> White, VertexLabelStyle -> Directive(Red, 15),
EdgeStyle -> Blue, VertexShapeFunction -> “Square”)


graph2 = ReverseGraph(graph1,VertexCoordinates -> GraphEmbedding(graph1))

However, this merely reverses the arrows. I need to take the mirror image of the graph (with respect to a horizontal line drawn above the graph) so that minimal elements end up at the bottom, maximal at the top.

So a ^ shaped Hasse diagram, should end up as a v shaped diagram. However nodes need to stay in the correct order: the vertices that were the minimal elements in ^ (the two bottom elements with arrows drawn down from the top element, i.e. minimum) should end up as minimal elements in the result (with arrows drawn up from the bottom element, i.e. minimum).

Is there an operation that achieves it?

Number of directed Hamiltonian paths in the complement of the $n$-path graph $P_n$

How do you calculate the number of directed Hamiltonian paths in the complement of the $n$-path graph $P_n$. Any help would be much appreciated, thanks

reference request – Graph theory from a category theory perspective

Are there any textbooks on graph theory written for a category theorist?

It would probably have to be on directed graph theory, but if there’s some trick we can use to talk about undirected graphs as well that would be interesting.

A little more specifically, I’m looking for a text that begins by defining directed graphs and paths, then defines the obvious category out of a given directed graph with paths as arrows, then proceeds to derive results about directed graphs using these categories.

Most connections I see made between category theory and graph theory are in the other direction, taking the underlying graph of a category and saying something about it to derive a result about the category, but as someone comfortable with categories and not comfortable with graphs this approach isn’t particularly illuminating.

Further, unless I’m mistaken, these constructions amount to an equivalence (maybe even an isomorphism?) between the category of directed graphs and the category of categories, so it feels like we should be able to say something about directed graphs from this perspective.

Any references are appreciated.

graph theory – Is there a purely set-theoretic expression of the Euler characteristic?

The Euler characteristic of a graph is V – E + F. Where V is the number of vertices, E is the number of edges, and F is the number of faces.

Given a graph $X$, can we write $X$ as a set and find its Euler characteristic with a formula which only refers to cardinality and union?

I think we found a way, but is this interesting? We would like to find a way to write the Euler characteristic completely set-theoretically. And, we have found a way, but still working on it. Has this already been done?

graph theory – Spanners defined via $K_4$ matchings

I have by accident found an interesting kind of spanner of complete symmetric graphs $G(V,E)$ with weighted edges.
What I actually had planned was to implement an algorithm for calculating certain non-optimal edges of TSP instances, but due to a bug I had actually calculated the set $lbrace a,b rbrace$ of edges with the property that for $forall cin Esetminuslbrace a,brbrace: exists din Esetminuslbrace a,b,crbrace $ s.t. $ omega_{ab}+omega_{cd}leomega_{ac}+omega_{bd},land,omega_{ab}+omega_{cd}leomega_{ad}+omega_{bc}$, i.e. for every vertex $c$ that is not adjacent to such an edge we can find another non-adjacent vertex $d$ such that the edges $lbrace a,brbrace$ and $lbrace c,drbrace$ resemble the minimum weight perfect matching of the subgraph induced by vertices $a,b,c,d$

Visualization of the K4 matching edges

The resulting graph is “almost” biconnected and may serve as the basis for shape hulls or partitioning of point sets. crossing pairs of edges are colored blue and the other, two-optimal subset of edges is depicted in yellow.


have spanners that are defined by the edges of certain $K_4$ matchings already been investigated, resp. can anything be said about their properties?

How would I graph this function on Mathematica?

enter image description here

I could graph 1/n^2, but I don’t understand how to graph the fractional part (x).

How to change my website’s map data image (maybe from knowledge graph) in the search results?

API Google Knowledge Graph scrapes structured data which in turn for images Google supports for the following types:

It is not clear from your question what type your image is.
You can use these Google guidelines for your images:

To non-amp: For best results, provide multiple high-resolution images
(minimum of 300,000 pixels when multiplying width and height) with the
following aspect ratios: 16×9, 4×3, and 1×1.

To amp: For best results, provide multiple high-resolution images
(minimum of 800,000 pixels when multiplying width and height) with the
following aspect ratios: 16×9, 4×3, and 1×1.

There is also an example there:

  "@context": "",
  "@type": "NewsArticle",
  "image": (

In addition, even if there is structured data consistent with Google’s guide, they do not undertake to use this data in a rich result. Check out the following Google statement about structured data:

Important: Google does not guarantee that your structured data will
show up in search results, even if your page is marked up correctly
according to the Rich Results Test. Here are some common reasons why:

Using structured data enables a feature to be present, it does not
guarantee that it will be present. The Google algorithm tailors search
results to create what it thinks is the best search experience for a
user, depending on many variables, including search history, location,
and device type. In some cases it may determine that one feature is
more appropriate than another, or even that a plain blue link is best.
The structured data is not representative of the main content of the
page, or is potentially misleading. The structured data is incorrect
in a way that the Rich Results Test was not able to catch. The content
referred to by the structured data is hidden from the user. The page
does not meet the guidelines for structured data described here, the
type-specific guidelines, or the general webmaster guidelines.

graph theory – A Question from Mathematical circles Book

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Microsoft Graph change notifications — For SharePoint Group for Tenant Level

I have requirement that whenever there will be any new user added in in SharePoint Site Group. I need to put the details for that group , User, Site details in another Data base.

Can We user Microsoft Graph Change Notification for it?

Is Microsoft Graph Change Notification support for SharePoint Group?

sharepoint framework – Fetching Planner Task through Graph API into webpart

public componentDidMount() {
this.props.context.msGraphClientFactory.getClient().then((client: MSGraphClient): void => {
    .get((error: any, eventsResponse, rawResponse?: any) => {
      if (error) {
        console.error("message is :" + error);
      const plannerTask: MicrosoftGraph.Event() = eventsResponse.value;
      this.setState({ events: plannerTask });

Making an API call through Graph Explorer to fetch the Planner Task data into sharepoint webpart in the above function

i am getting an error when fetching the data of “{“Assignments : AssignedBy :User :id”}”

I am fetching data through Graph Explorer as shown below

enter image description here

I am able to fetch the {“CreatedBy : User :id”} but not able to fetch {“Assignments : AssignedBy :User :id”}

I have used that code to fetch the createdBy’s ID and assignedBY’s ID

{, key) =>
              <td id="status">
                {(() => {
                  switch (item.percentComplete) {
                    case 0: return "Not-started";
                    case 50: return "In-Progress";
                    case 100: return "Completed";
              <td><a href={"" + item.planId + "&taskId=" +} target="_blank" >{item.title}</a></td>
              <td>{moment(item.startDateTime).format('MM/DD/YYYY ')}</td>