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We study the set\n                    <jats:italic>S<\/jats:italic>\n                    (\n                    <jats:italic>G<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>c<\/jats:italic>\n                    ) of connected subgraph sums and, in particular, resolve a problem posed by O.-H. S. Lo in a strong form. We show that for each\n                    <jats:italic>n<\/jats:italic>\n                    -vertex graph\n                    <jats:italic>G<\/jats:italic>\n                    , there is a vertex assignment\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$c:V(G)\\rightarrow \\{1,\\dots ,12n^2\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mo>\u22ef<\/mml:mo>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>12<\/mml:mn>\n                              <mml:msup>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that for every\n                    <jats:italic>n<\/jats:italic>\n                    -vertex graph\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$G'\\not \\cong G$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo>\u2247<\/mml:mo>\n                            <mml:mi>G<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and vertex assignment\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$c'$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>\u2032<\/mml:mo>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$G'$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>\u2032<\/mml:mo>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the corresponding collections of connected subgraph sums are different (i.e.,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$S(G,c)\\ne S(G',c')$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>G<\/mml:mi>\n                                <mml:mo>\u2032<\/mml:mo>\n                              <\/mml:msup>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>c<\/mml:mi>\n                                <mml:mo>\u2032<\/mml:mo>\n                              <\/mml:msup>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ). We also provide some remarks on vertex assignments of a graph\n                    <jats:italic>G<\/jats:italic>\n                    for which all connected subgraph sums are different.\n                  <\/jats:p>","DOI":"10.1007\/s00373-026-03026-8","type":"journal-article","created":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T03:28:53Z","timestamp":1772767733000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Note on extremal problems about connected subgraph sums"],"prefix":"10.1007","volume":"42","author":[{"given":"Stijn","family":"Cambie","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9878-8750","authenticated-orcid":false,"given":"Carla","family":"Groenland","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,6]]},"reference":[{"key":"3026_CR1","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ejc.2023.103712","volume":"112","author":"M Axenovich","year":"2023","unstructured":"Axenovich, M., Caro, Y., Yuster, R.: Sum-distinguishing number of sparse hypergraphs. 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