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On the Maximum Number of Spanning Trees in $C_4$-Free Graphs

András London

Abstract

We introduce a ``Kirchhoff--Turán'' variant of the extremal $C_4$ problem: among all simple connected $n$-vertex $C_4$-free graphs $G$, maximize the number of spanning trees $τ(G)$. For the projective-plane orders $n=q^2+q+1$ we compute an exact formula for the Erdős--Rényi orthogonal polarity graph $ER_q$, namely $τ(ER_q)=n^{(n-3)/2}$, via a polarity spectral identity and Kirchhoff's matrix--tree theorem. We also give an explicit general upper bound on $\mathrm{st}(n,C_4)$ at these $n$ using a sharp degree-sequence inequality for $τ(G)$ and a degree-balancing argument; this matches the lower bound in the leading exponential term.

On the Maximum Number of Spanning Trees in $C_4$-Free Graphs

Abstract

We introduce a ``Kirchhoff--Turán'' variant of the extremal problem: among all simple connected -vertex -free graphs , maximize the number of spanning trees . For the projective-plane orders we compute an exact formula for the Erdős--Rényi orthogonal polarity graph , namely , via a polarity spectral identity and Kirchhoff's matrix--tree theorem. We also give an explicit general upper bound on at these using a sharp degree-sequence inequality for and a degree-balancing argument; this matches the lower bound in the leading exponential term.
Paper Structure (11 sections, 10 theorems, 32 equations)

This paper contains 11 sections, 10 theorems, 32 equations.

Table of Contents

  1. The spanning-tree Turán problem for $C_4$
  2. Warm-up: the triangle-free case.
  3. Two ingredients: a degree bound and degree balancing
  4. Orthogonal polarity graphs and an exact spanning-tree count
  5. An explicit upper bound at $n=q^2+q+1$
  6. A quantitative deficit bound in a near-extremal regime
  7. Future directions
  8. The case of general $n$
  9. Forbidding $K_{2,t}$ and $C_{2k}$
  10. The family $K_{2,t}$
  11. Even cycles $C_{2k}$

Key Result

Theorem 1

For every simple graph $G$ on $n$ vertices,

Theorems & Definitions (26)

  • Definition 1
  • Remark 1
  • Theorem 1: Klee--Narayanan--Sauermann KNS-arxiv
  • Remark 2
  • Lemma 1: Balancing
  • proof
  • Theorem 2: Füredi; see e.g. MaYangBLMS23HeMaYangCSIAM23
  • Definition 2: Orthogonal polarity graph $ER_q$
  • Remark 3: Degrees
  • Lemma 2: Spectrum of polarity graphs; multiplicities from absolute points TaitTimmonsAJC15
  • ...and 16 more