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The pebbling number of a graph <jats:italic>G<\/jats:italic> is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of <jats:italic>G<\/jats:italic> through some sequence of pebbling moves. Through constructing a non-tree weight function for <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$Q_4$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Q<\/mml:mi>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the <jats:italic>n<\/jats:italic>-dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions.<\/jats:p>","DOI":"10.1007\/s10878-024-01248-1","type":"journal-article","created":{"date-parts":[[2024,12,25]],"date-time":"2024-12-25T13:01:16Z","timestamp":1735131676000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lollipop and cubic weight functions for graph pebbling"],"prefix":"10.1007","volume":"49","author":[{"given":"Marshall","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carl","family":"Yerger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-7525-3541","authenticated-orcid":false,"given":"Runtian","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,25]]},"reference":[{"issue":"3","key":"1248_CR1","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1002\/jgt.20278","volume":"57","author":"DP Bunde","year":"2008","unstructured":"Bunde DP, Chambers EW, Cranston D, Milans K, West DB (2008) Pebbling and optimal pebbling in graphs. 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