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

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













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.







$endgroup$
















    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.







    $endgroup$














      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






      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









      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




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









      asked 5 hours ago









      Ishaan ShahIshaan Shah

      264




      264




      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.





      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.






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




















          2 Answers
          2






          active

          oldest

          votes


















          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 $fracrn2$ 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 $(1,frac3227]$ ) 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











          $endgroup$




















            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









            $endgroup$













              Your Answer





              StackExchange.ifUsing("editor", function ()
              return StackExchange.using("mathjaxEditing", function ()
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              );
              );
              , "mathjax-editing");

              StackExchange.ready(function()
              var channelOptions =
              tags: "".split(" "),
              id: "504"
              ;
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function()
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled)
              StackExchange.using("snippets", function()
              createEditor();
              );

              else
              createEditor();

              );

              function createEditor()
              StackExchange.prepareEditor(
              heartbeatType: 'answer',
              autoActivateHeartbeat: false,
              convertImagesToLinks: true,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              imageUploader:
              brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
              contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
              allowUrls: true
              ,
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              );



              );






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









              draft saved

              draft discarded


















              StackExchange.ready(
              function ()
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327426%2fwhat-makes-graph-invariants-so-useful-important%23new-answer', 'question_page');

              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              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 $fracrn2$ 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 $(1,frac3227]$ ) 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 $fracrn2$ 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 $(1,frac3227]$ ) 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 $fracrn2$ 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 $(1,frac3227]$ ) 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 $fracrn2$ 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 $(1,frac3227]$ ) 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 1 hour 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.









                              draft saved

                              draft discarded


















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












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











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














                              Thanks for contributing an answer to MathOverflow!


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid


                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.

                              Use MathJax to format equations. MathJax reference.


                              To learn more, see our tips on writing great answers.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function ()
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327426%2fwhat-makes-graph-invariants-so-useful-important%23new-answer', 'question_page');

                              );

                              Post as a guest















                              Required, but never shown





















































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown

































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown







                              Popular posts from this blog

                              名間水力發電廠 目录 沿革 設施 鄰近設施 註釋 外部連結 导航菜单23°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.7113923°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.71139計畫概要原始内容臺灣第一座BOT 模式開發的水力發電廠-名間水力電廠名間水力發電廠 水利署首件BOT案原始内容《小檔案》名間電廠 首座BOT水力發電廠原始内容名間電廠BOT - 經濟部水利署中區水資源局

                              Prove that NP is closed under karp reduction?Space(n) not closed under Karp reductions - what about NTime(n)?Class P is closed under rotation?Prove or disprove that $NL$ is closed under polynomial many-one reductions$mathbfNC_2$ is closed under log-space reductionOn Karp reductionwhen can I know if a class (complexity) is closed under reduction (cook/karp)Check if class $PSPACE$ is closed under polyonomially space reductionIs NPSPACE also closed under polynomial-time reduction and under log-space reduction?Prove PSPACE is closed under complement?Prove PSPACE is closed under union?

                              Is my guitar’s action too high? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Strings too stiff on a recently purchased acoustic guitar | Cort AD880CEIs the action of my guitar really high?Μy little finger is too weak to play guitarWith guitar, how long should I give my fingers to strengthen / callous?When playing a fret the guitar sounds mutedPlaying (Barre) chords up the guitar neckI think my guitar strings are wound too tight and I can't play barre chordsF barre chord on an SG guitarHow to find to the right strings of a barre chord by feel?High action on higher fret on my steel acoustic guitar