We construct a weighted signed dual multigraph for the Cayley $2$-complex of a finite presentation and represent its $\ell^2$ face Laplacian as a signed graph Laplacian with a diagonal potential. We prove that the face spectral gap above the kernel vanishes for the Baumslag--Solitar groups $\operatorname{BS}(1,m)$, $m\ge2$, using explicit finitely supported face chains. Our main vanishing theorem gives the same conclusion for every finitely presented amenable group with ordinary first Betti number at least two.
For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian $\Delta$, we prove weak type $(1,1)$ estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carr\'e-du-champ transform has constant $H...
Let $\Ga$ be a connected prime-valent $X$-arc-transitive Cayley graph of a finite characteristically simple group $G\cong T^k$, where $k\geqslant2$. We obtain a precise structural characterization of such graphs and their arc-transitive automorphism groups. In the cubic case, every connected symmetric Cayley graph of $...
We introduce the concept of parameterized twisted Seidel energy by incorporating a real parameter $\alpha$ into the Seidel matrix definition. This generalization yields a family of graph energies that interpolate between classical Seidel energy ($\alpha = 0$) and complementary Seidel energy ($\alpha = 1$). We establish...
We study a Calder\'on-type inverse problem for the spectral fractional Laplacian on finite Cayley graphs. For a known self-adjoint operator, recovery from a single measurement reduces to two requirements: full rank of the boundary (exterior)-to-interior coupling and nonvanishing of the interior solution. We verify both...
Let $\mathcal F=\{2K_2,C_4,C_5\}$, so that $\operatorname{Forb}(\mathcal F)$ is the class of split graphs. Resolving a conjecture of Wang and Tang, we prove that the class of split graphs is generalized spectrally closed and that it admits no walk-realizable $\mathcal F$-supporter of any finite order.
In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen--Franks groups associated with these graphs in several...
Debanjana Kundu, Katharina Müller, A. Newton et al.· 0 citations
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