What makes Graph invariants so useful/important?Complete graph invariants?A Combinatorial Abstraction for The...



What makes Graph invariants so useful/important?


Complete graph invariants?A Combinatorial Abstraction for The “Polynomial Hirsch Conjecture” What graph invariants are fast to compute?What are some useful invariants for distinguishing between random graph models?Graph homomorphisms and line graphCalculating a Combinatorial Generalization of Planar Convex HullsOn the use of Weisfeiler-Leman refinement in Babai's GI proofReconstruction conjecture:Complete graph invariantsChromatic number of the plane and phase transitions of Potts modelsGraph isomorphism by invariants













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$begingroup$


What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow?



My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canonical labellings are the right tool for this).



However, important graph invariants are constructed from counting proper colorings of a graph, for an appropriate definition of proper. A priori, why do we know that those graph invariants isolate and study specific properties or is there some other key motivation for graph invariants?










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    $begingroup$


    What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow?



    My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canonical labellings are the right tool for this).



    However, important graph invariants are constructed from counting proper colorings of a graph, for an appropriate definition of proper. A priori, why do we know that those graph invariants isolate and study specific properties or is there some other key motivation for graph invariants?










    share|cite|improve this question









    New contributor




    Ishaan Shah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$















      3












      3








      3





      $begingroup$


      What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow?



      My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canonical labellings are the right tool for this).



      However, important graph invariants are constructed from counting proper colorings of a graph, for an appropriate definition of proper. A priori, why do we know that those graph invariants isolate and study specific properties or is there some other key motivation for graph invariants?










      share|cite|improve this question









      New contributor




      Ishaan Shah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow?



      My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canonical labellings are the right tool for this).



      However, important graph invariants are constructed from counting proper colorings of a graph, for an appropriate definition of proper. A priori, why do we know that those graph invariants isolate and study specific properties or is there some other key motivation for graph invariants?







      co.combinatorics graph-theory graph-colorings






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      New contributor




      Ishaan Shah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      Ishaan Shah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago







      Ishaan Shah













      New contributor




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      asked 5 hours ago









      Ishaan ShahIshaan Shah

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      New contributor





      Ishaan Shah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






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          $begingroup$

          We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$



          If you want to decide “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.



          The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.



          As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.



          You might start with a question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.



          In some cases we also might want to know if there are or could be graphs which have a certain mix of invariants such as regular of degree $r$ with $n$ vertices. Then the invariant “number of edges” is $frac{rn}2$ which tells you that $r$ and $n$ can’t both be odd. That is a basic example but there are spectacular results obtained by considering the distinct eigenvalues and multiplicities (a negative or non-integer multiplicity says “no way!”)



          LATER Since you ask, here is an example of my claim that the invariant arises from a question. Map coloring leads to Problem: show that every planar graph enjoys the (invariant) property of having a proper $4$-coloring. This leads the one to define the chromatic function $P(G,c)$ as the number of proper colorings of the given graph $G$ with $c$ colors to show $P(G,4) gt 0$ for planar graphs. Once you start to investigate there is the perhaps unexpected discovery that $P(G,c)=p(c)$ for a polynomial $p=p_G(x).$ By that time the chromatic polynomial seems well motivated and finding that $p(-1)$ counts acyclic orientations just ups the ante. Read Wikipedia for details.



          There are intervals of the real line which can not contain any zeroes of a chromatic polynomial ( for example $(0,1)$ and $(1,frac{32}{27}]$ ) and also some intervals which can’t do so for a planar triangulation of a sphere. Alas, no one has shown algebraically that $4$ belongs to such an interval.






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            0












            $begingroup$

            Canonical labellings are hard to find and to handle. This is why one neeeds invariants, so that one can have statements like “all graphs having such and such invariants are so and so”.



            This is akin to many other mathematical theories, e.g. a lot can be said about a linear operator from its characteristic polynomial alone.
            By the way, the characteristic polynomial of the adjacency matrix of a graph contains quite a bit of information about the graph, e.g. in some cases one can say things about the diameter of the graph, etc.






            share|cite|improve this answer









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              2 Answers
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              $begingroup$

              We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$



              If you want to decide “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.



              The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.



              As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.



              You might start with a question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.



              In some cases we also might want to know if there are or could be graphs which have a certain mix of invariants such as regular of degree $r$ with $n$ vertices. Then the invariant “number of edges” is $frac{rn}2$ which tells you that $r$ and $n$ can’t both be odd. That is a basic example but there are spectacular results obtained by considering the distinct eigenvalues and multiplicities (a negative or non-integer multiplicity says “no way!”)



              LATER Since you ask, here is an example of my claim that the invariant arises from a question. Map coloring leads to Problem: show that every planar graph enjoys the (invariant) property of having a proper $4$-coloring. This leads the one to define the chromatic function $P(G,c)$ as the number of proper colorings of the given graph $G$ with $c$ colors to show $P(G,4) gt 0$ for planar graphs. Once you start to investigate there is the perhaps unexpected discovery that $P(G,c)=p(c)$ for a polynomial $p=p_G(x).$ By that time the chromatic polynomial seems well motivated and finding that $p(-1)$ counts acyclic orientations just ups the ante. Read Wikipedia for details.



              There are intervals of the real line which can not contain any zeroes of a chromatic polynomial ( for example $(0,1)$ and $(1,frac{32}{27}]$ ) and also some intervals which can’t do so for a planar triangulation of a sphere. Alas, no one has shown algebraically that $4$ belongs to such an interval.






              share|cite|improve this answer











              $endgroup$


















                4












                $begingroup$

                We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$



                If you want to decide “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.



                The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.



                As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.



                You might start with a question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.



                In some cases we also might want to know if there are or could be graphs which have a certain mix of invariants such as regular of degree $r$ with $n$ vertices. Then the invariant “number of edges” is $frac{rn}2$ which tells you that $r$ and $n$ can’t both be odd. That is a basic example but there are spectacular results obtained by considering the distinct eigenvalues and multiplicities (a negative or non-integer multiplicity says “no way!”)



                LATER Since you ask, here is an example of my claim that the invariant arises from a question. Map coloring leads to Problem: show that every planar graph enjoys the (invariant) property of having a proper $4$-coloring. This leads the one to define the chromatic function $P(G,c)$ as the number of proper colorings of the given graph $G$ with $c$ colors to show $P(G,4) gt 0$ for planar graphs. Once you start to investigate there is the perhaps unexpected discovery that $P(G,c)=p(c)$ for a polynomial $p=p_G(x).$ By that time the chromatic polynomial seems well motivated and finding that $p(-1)$ counts acyclic orientations just ups the ante. Read Wikipedia for details.



                There are intervals of the real line which can not contain any zeroes of a chromatic polynomial ( for example $(0,1)$ and $(1,frac{32}{27}]$ ) and also some intervals which can’t do so for a planar triangulation of a sphere. Alas, no one has shown algebraically that $4$ belongs to such an interval.






                share|cite|improve this answer











                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$



                  If you want to decide “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.



                  The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.



                  As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.



                  You might start with a question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.



                  In some cases we also might want to know if there are or could be graphs which have a certain mix of invariants such as regular of degree $r$ with $n$ vertices. Then the invariant “number of edges” is $frac{rn}2$ which tells you that $r$ and $n$ can’t both be odd. That is a basic example but there are spectacular results obtained by considering the distinct eigenvalues and multiplicities (a negative or non-integer multiplicity says “no way!”)



                  LATER Since you ask, here is an example of my claim that the invariant arises from a question. Map coloring leads to Problem: show that every planar graph enjoys the (invariant) property of having a proper $4$-coloring. This leads the one to define the chromatic function $P(G,c)$ as the number of proper colorings of the given graph $G$ with $c$ colors to show $P(G,4) gt 0$ for planar graphs. Once you start to investigate there is the perhaps unexpected discovery that $P(G,c)=p(c)$ for a polynomial $p=p_G(x).$ By that time the chromatic polynomial seems well motivated and finding that $p(-1)$ counts acyclic orientations just ups the ante. Read Wikipedia for details.



                  There are intervals of the real line which can not contain any zeroes of a chromatic polynomial ( for example $(0,1)$ and $(1,frac{32}{27}]$ ) and also some intervals which can’t do so for a planar triangulation of a sphere. Alas, no one has shown algebraically that $4$ belongs to such an interval.






                  share|cite|improve this answer











                  $endgroup$



                  We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$



                  If you want to decide “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.



                  The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.



                  As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.



                  You might start with a question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.



                  In some cases we also might want to know if there are or could be graphs which have a certain mix of invariants such as regular of degree $r$ with $n$ vertices. Then the invariant “number of edges” is $frac{rn}2$ which tells you that $r$ and $n$ can’t both be odd. That is a basic example but there are spectacular results obtained by considering the distinct eigenvalues and multiplicities (a negative or non-integer multiplicity says “no way!”)



                  LATER Since you ask, here is an example of my claim that the invariant arises from a question. Map coloring leads to Problem: show that every planar graph enjoys the (invariant) property of having a proper $4$-coloring. This leads the one to define the chromatic function $P(G,c)$ as the number of proper colorings of the given graph $G$ with $c$ colors to show $P(G,4) gt 0$ for planar graphs. Once you start to investigate there is the perhaps unexpected discovery that $P(G,c)=p(c)$ for a polynomial $p=p_G(x).$ By that time the chromatic polynomial seems well motivated and finding that $p(-1)$ counts acyclic orientations just ups the ante. Read Wikipedia for details.



                  There are intervals of the real line which can not contain any zeroes of a chromatic polynomial ( for example $(0,1)$ and $(1,frac{32}{27}]$ ) and also some intervals which can’t do so for a planar triangulation of a sphere. Alas, no one has shown algebraically that $4$ belongs to such an interval.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 57 mins ago

























                  answered 3 hours ago









                  Aaron MeyerowitzAaron Meyerowitz

                  24.4k13388




                  24.4k13388























                      0












                      $begingroup$

                      Canonical labellings are hard to find and to handle. This is why one neeeds invariants, so that one can have statements like “all graphs having such and such invariants are so and so”.



                      This is akin to many other mathematical theories, e.g. a lot can be said about a linear operator from its characteristic polynomial alone.
                      By the way, the characteristic polynomial of the adjacency matrix of a graph contains quite a bit of information about the graph, e.g. in some cases one can say things about the diameter of the graph, etc.






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        Canonical labellings are hard to find and to handle. This is why one neeeds invariants, so that one can have statements like “all graphs having such and such invariants are so and so”.



                        This is akin to many other mathematical theories, e.g. a lot can be said about a linear operator from its characteristic polynomial alone.
                        By the way, the characteristic polynomial of the adjacency matrix of a graph contains quite a bit of information about the graph, e.g. in some cases one can say things about the diameter of the graph, etc.






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          Canonical labellings are hard to find and to handle. This is why one neeeds invariants, so that one can have statements like “all graphs having such and such invariants are so and so”.



                          This is akin to many other mathematical theories, e.g. a lot can be said about a linear operator from its characteristic polynomial alone.
                          By the way, the characteristic polynomial of the adjacency matrix of a graph contains quite a bit of information about the graph, e.g. in some cases one can say things about the diameter of the graph, etc.






                          share|cite|improve this answer









                          $endgroup$



                          Canonical labellings are hard to find and to handle. This is why one neeeds invariants, so that one can have statements like “all graphs having such and such invariants are so and so”.



                          This is akin to many other mathematical theories, e.g. a lot can be said about a linear operator from its characteristic polynomial alone.
                          By the way, the characteristic polynomial of the adjacency matrix of a graph contains quite a bit of information about the graph, e.g. in some cases one can say things about the diameter of the graph, etc.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 5 hours ago









                          Dima PasechnikDima Pasechnik

                          9,38311952




                          9,38311952






















                              Ishaan Shah is a new contributor. Be nice, and check out our Code of Conduct.










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