# Adjective for objects (graphs, networks or polygons) having the same number of edges

I am looking for an adjective form to describe similar geometric objects (graphs, networks, polygons) that have "the same number of edges" (for a technical audience, a scientific paper). [EDIT] Based on comments, I should add that I understand polygons as possibly open: a square without one edge (U-shaped) possesses the same number of edges as a closed triangle. I am more informated in a adjective related to a count than to a shape ("morhp-").

This expression ("having the same number of edges") occurs several times, and I would like to have variations. A typical sentence would be:

We thus compare two networks with the same number of edges

Starting from equal-sized and the like, I thought about "equally-edged polygons" but this does not sound correct to me. Indeed, a graph can be described by different sizes: the number of edges, and the number of vertices. So equal-sized is ambiguous.

I would like a construction that could also apply on "having the same number of vertices" or "having the same number of facets" if possible.

• Hint: Isomorphic - being of identical or similar form, shape, or structure – MorganFR May 13 '16 at 12:35
• @MorganFR: Isomorphic is wrong. There is a technical definition of isomorphic in mathematics, and two graphs with the same number of edges need not be isomorphic. – Peter Shor May 13 '16 at 12:59
• It was only a hint. – MorganFR May 13 '16 at 13:33
• All polygons have specific names; you merely need a synonymy for same. If you want words to replace "having the same number of vertices/facets" then you need an example sentence. – Mazura May 13 '16 at 15:03
• @Mazura The example osentence has been added – Laurent Duval May 13 '16 at 22:10

You could use the term edge count. It's slightly more compact than "number of edges".

We thus compare two networks with the same edge count.

We thus compare two networks that have equal edge counts.

Edge Count

The edge count of a graph g, commonly denoted M(g) or E(g) and sometimes also called the edge number, is the number of edges in g. In other words, it is the cardinality of the edge set.

• Sounds good to me, and can generalize to vertex-count and similar. – Laurent Duval Jun 13 '16 at 7:59
• @LaurentDuval, I've updated my answer with a reference, and I've also remove the unnecessary hyphen from edge count. Note also that Wolfram says you can call it M(g) or E(g) for some graph g. – dangph Jun 13 '16 at 8:22

Isomorphic meaning having the same form is a possible answer, as long as this is not taken to imply "having the same size" when applied to lengths.

I can imagine in referring to a collection of polygons of different numbers of sides and different sizes, that one could say "color all isomorphic polygons in one color, using different colors for different shapes". Here isomorphic would look like over-kill. But if you add polyhedra to the mix, and then say "all isomorphic shapes" or even "all isomorphs", you will be economical of your words.

• In mathematics, isomorphic has a technical definition, and while for polygons, it does mean same number of edges, for graphs and polyhedra, it does not. – Peter Shor May 13 '16 at 13:00
• That the word has a technical definition in mathematics does not preclude its use in the present case. I can think of lots of words with strict mathematical definitions that are used in ways far from their definition in English:* square, line, angle, circle, sector * come to mind immediately. – frank May 13 '16 at 13:07
• That depends on what the OP wants the word for. If he's writing something for a technical audience, it's both technically wrong and misleading. And if he's not writing something for a technical audience, why is he using vertices and facets rather than corners and faces? – Peter Shor May 13 '16 at 13:08
• He didn't say, but you could ask him. – frank May 13 '16 at 13:11

Where n-gon refers to any number of sides, x-gon refers to some fixed number of sides. "x" is understood to be single valued throughout. It works for most things, but I'd explain it once in the article anyway.

isomorphism, which has already been suggested, which means

a one-to-one correspondence between two mathematical sets; especially :  a homomorphism that is one-to-one.

• -1: Graph isomorphism (did you read your link‽‽) does not mean "same number of edges". – Peter Shor May 13 '16 at 13:01
• ... and isomorphism when applied to graphs would automatically be interpreted by mathematicians and computer scientists as graph isomorphism. This is not the right word. – Peter Shor May 13 '16 at 13:06
• @PeterShor what about polygons? – vickyace May 13 '16 at 13:08
• You can use it for polygons. – Peter Shor May 13 '16 at 13:10
• I am not sure isomorphism would work for crossed polygons, due to the change in topology – Laurent Duval May 13 '16 at 13:22