the mean subtree order and the mean connected induced
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B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK The Mean Subtree Order and the Mean Connected Induced Subgraph Order Lucas Mol Joint work with Kristaps Balodis and Ortrud Oellermann (The University of Winnipeg), and Matthew Kroeker (The

  1. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 15:

  2. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 16:

  3. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 17:

  4. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 18:

  5. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 19:

  6. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 20:

  7. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 21:

  8. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 22:

  9. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 23:

  10. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P ROGRESS ◮ Jamison asked six questions at the end of his 1983 paper. ◮ Five of them have recently been answered. The last one: ◮ Conjecture: The tree of order n with maximum mean subtree order is a caterpillar. Order 24:

  11. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P LAN B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK

  12. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK T HE G LUING L EMMA Let T be a tree of order at least 2 with vertex v . T v u 1 u 2 u k u n − 1 u n

  13. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK T HE G LUING L EMMA Let T be a tree of order at least 2 with vertex v . T v u 1 u 2 u k u n − 1 u n

  14. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK T HE G LUING L EMMA Let T be a tree of order at least 2 with vertex v . T u 1 u 2 u n − 1 u n u k = v

  15. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK T HE G LUING L EMMA Let T be a tree of order at least 2 with vertex v . T u 1 u 2 u n − 1 u n u k = v Call this tree T k .

  16. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK T HE G LUING L EMMA Let T be a tree of order at least 2 with vertex v . T u 1 u 2 u n − 1 u n u k = v Call this tree T k . The Gluing Lemma (Mol and Oellermann, 2018): If 1 ≤ r < s ≤ n + 1 2 , then M T r < M T s .

  17. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA

  18. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order.

  19. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  20. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  21. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  22. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  23. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  24. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  25. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  26. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  27. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  28. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (First proven by Jamison, 1983): Among all trees of order n , the path P n has minimum mean subtree order. Proof Idea: Recursively apply the Gluing Lemma.

  29. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (Mol and Oellermann, 2018): Let n ≥ 4, and suppose that T has the largest mean subtree order among all trees of order n . Then every leaf of T is adjacent to a vertex of degree at least 3.

  30. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (Mol and Oellermann, 2018): Let n ≥ 4, and suppose that T has the largest mean subtree order among all trees of order n . Then every leaf of T is adjacent to a vertex of degree at least 3. Proof: Suppose otherwise that T has a leaf u adjacent to a vertex of degree 2.

  31. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (Mol and Oellermann, 2018): Let n ≥ 4, and suppose that T has the largest mean subtree order among all trees of order n . Then every leaf of T is adjacent to a vertex of degree at least 3. Proof: Suppose otherwise that T has a leaf u adjacent to a vertex of degree 2. u

  32. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (Mol and Oellermann, 2018): Let n ≥ 4, and suppose that T has the largest mean subtree order among all trees of order n . Then every leaf of T is adjacent to a vertex of degree at least 3. Proof: Suppose otherwise that T has a leaf u adjacent to a vertex of degree 2.

  33. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK C OROLLARIES OF THE G LUING L EMMA Theorem (Mol and Oellermann, 2018): Let n ≥ 4, and suppose that T has the largest mean subtree order among all trees of order n . Then every leaf of T is adjacent to a vertex of degree at least 3. Proof: Suppose otherwise that T has a leaf u adjacent to a vertex of degree 2. . . .

  34. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK P LAN B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK

  35. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ?

  36. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G .

  37. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018).

  38. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018). 2. The mean order of all connected induced subgraphs of G .

  39. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018). 2. The mean order of all connected induced subgraphs of G . ◮ Considered by Balodis, Kroeker, Mol, and Oellermann (2018, 2019+).

  40. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018). 2. The mean order of all connected induced subgraphs of G . ◮ Considered by Balodis, Kroeker, Mol, and Oellermann (2018, 2019+). 3. The mean order of all connected (not necessarily induced) subgraphs of G .

  41. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018). 2. The mean order of all connected induced subgraphs of G . ◮ Considered by Balodis, Kroeker, Mol, and Oellermann (2018, 2019+). 3. The mean order of all connected (not necessarily induced) subgraphs of G . ◮ Not yet considered, but maybe interesting?

  42. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK B EYOND T REES How can we extend the mean subtree order to a general connected graph G ? 1. The mean order of all “subtrees” (i.e., trees that are subgraphs) of G . ◮ Considered by Chin, Gordon, MacPhee, and Vincent (2018). 2. The mean order of all connected induced subgraphs of G . ◮ Considered by Balodis, Kroeker, Mol, and Oellermann (2018, 2019+). 3. The mean order of all connected (not necessarily induced) subgraphs of G . ◮ Not yet considered, but maybe interesting? From now on, M G and M G , v denote the global and local versions, respectively, of the mean connected induced subgraph order (mean CIS order).

  43. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail!

  44. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): T . . . v

  45. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . T . . . v

  46. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . . . . v

  47. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . ◮ Call the resulting graph G . . . . v

  48. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . ◮ Call the resulting graph G . ◮ M G , v = n + 2 . . . < M G − v = M T . 2 v

  49. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . ◮ Call the resulting graph G . ◮ M G , v = n + 2 . . . < M G − v = M T . 2 ◮ M G is a weighted average of M G , v and v M G − v , so

  50. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . ◮ Call the resulting graph G . ◮ M G , v = n + 2 . . . < M G − v = M T . 2 ◮ M G is a weighted average of M G , v and v M G − v , so M G > M G , v .

  51. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A C HALLENGE The local/global mean inequality ( M G , v > M G ) can fail! Counterexample (Kroeker, Mol, and Oellermann, 2018): ◮ Let T be a tree of order n with M T > n + 2 2 . ◮ Add a new vertex v and join it to every T vertex of T . ◮ Call the resulting graph G . ◮ M G , v = n + 2 . . . < M G − v = M T . 2 ◮ M G is a weighted average of M G , v and v M G − v , so M G > M G , v . Challenge: Many important results for trees rely on the local/global mean inequality!

  52. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A R AY OF H OPE Andriantiana, Misanantenaina, and Wagner (2018): Proved in a more general setting that for any graph G of order at least 2, there exists a vertex v of G such that M G , v > M G .

  53. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A R AY OF H OPE Andriantiana, Misanantenaina, and Wagner (2018): Proved in a more general setting that for any graph G of order at least 2, there exists a vertex v of G such that M G , v > M G . Questions:

  54. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A R AY OF H OPE Andriantiana, Misanantenaina, and Wagner (2018): Proved in a more general setting that for any graph G of order at least 2, there exists a vertex v of G such that M G , v > M G . Questions: ◮ For which graphs does the local/global mean inequality still hold at every vertex?

  55. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK A R AY OF H OPE Andriantiana, Misanantenaina, and Wagner (2018): Proved in a more general setting that for any graph G of order at least 2, there exists a vertex v of G such that M G , v > M G . Questions: ◮ For which graphs does the local/global mean inequality still hold at every vertex? ◮ At what proportion of vertices can the local/global mean inequality fail?

  56. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order.

  57. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 4:

  58. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 3:

  59. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 4:

  60. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 5:

  61. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 6:

  62. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 7:

  63. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 8:

  64. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK E XTREMAL G RAPHS Conjectured Minimum: Among all connected graphs of order n , the path P n has smallest mean CIS order. Maximum: Hard to tell what to conjecture. Order 9:

  65. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK G RAPH C LASSES The problem in general seems really difficult! What can we say about graph classes?

  66. B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK G RAPH C LASSES The problem in general seems really difficult! What can we say about graph classes? ◮ Cographs: Kroeker, Mol, and Oellermann, 2018

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