If I use $(W(E), V, E)$ to denote a graph with $E$ set of edges $V$ set of vertices and edge weight $W(E)$. Then, will my notations ($\emptyset, \infty, \emptyset$) and ($\emptyset, \emptyset, \emptyset$) be appropriate to denote an empty graph and a null graph, respectively? Where empty graph is defined as the graph with no edges and no vertices and null graph is defined as a graph with no edges as defined by an expert here.

  • $\begingroup$ I think the null graph is "a" graph without edges. So I would use some non empty set $S$ instead of $\infty$. Is $\infty$ even a set in your context? $\endgroup$ – SK19 Feb 17 '18 at 7:28
  • $\begingroup$ Yes, i consider infinity as a possible set that may contain infinitely many isolated vertices. And V is an arbitrary set containing connected vertices $\endgroup$ – gete Feb 17 '18 at 7:57

The infinity symbol plays something of a different role in graph theory and I would not intuitively expect it to denote an arbitrary set of vertices. This confusion is compounded by the fact that you already have a symbol for an arbitrary vertex set $V$. So unless your use of $\infty$ were clarified I would consider this poor notation.

  • $\begingroup$ I made a slight change in the sentence above , kindly note, sir. Here, i consider infinity as a set that contains infinitely many isolated vertices . While V is a set of connected vertices. $\endgroup$ – gete Feb 17 '18 at 8:04
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    $\begingroup$ @gete $V$ and "$\infty$" are sets you consider before defining the graph, therefore it doesn't make much sense to differ between them by saying one contains only "isolated" and the other contains only "connected" vertices. It would also be questionable what should be used for the set when dealing with graphs where some but not all vertices are isolated. $\endgroup$ – SK19 Feb 17 '18 at 8:19
  • $\begingroup$ Sir, i have consider such notations in trying to make the notations acts as identity element with respect to certain binary operations when we would like to make the collection of graphs under binary operations forms some algebraic structure. Thank you sir for your concern $\endgroup$ – gete Feb 17 '18 at 8:46

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