Related papers: Iwasawa theory for vertex-weighted graphs
Let $p$ be a prime number and let $d$ be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Valli\`{e}res to weighted graphs. In particular, we prove an analogue of Iwasawa's class number…
We initiate the study of Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite connected graphs. These towers are more general than what have been studied so far, since the morphisms of graphs involved are branched covers, a…
We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we…
This paper explores Iwasawa theory from a graph theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within…
Recently Iwasawa theory for graphs is developing. A significant achievement includes an analogue of Iwasawa class number formula, which describes the asymptotic growth of the numbers of spanning trees for $\mathbb{Z}_p$-coverings of graphs.…
A graph-theoretic analogue of Iwasawa theory, initiated by Gonet and Valli\`eres, has attracted considerable interest in the study of Iwasawa invariants. On the other hand, for a pair of prime numbers $(r,\ell)$, one obtains a graph, called…
In this note, I develop a representation-theoretic refinement of the Iwasawa theory of finite Cayley graphs. Building on analogies between graph zeta functions and number-theoretic L-functions, I study $\mathbb{Z}_\ell$-towers of Cayley…
Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the…
In this article, we study questions pertaining to ramified $\mathbb{Z}_p^d$-extensions of a finite connected graph $X$. We also study the Iwasawa theory of dual graphs.
The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\textit{J. Combin. Theory Ser. B}, 74 (1998), pp.…
We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion…
Let $\ell$ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell$-towers of multigraphs. In this context, growth patterns are realized by certain…
Gr\"otschel, Lov\'asz, and Schrijver generalized the Lov\'asz $\vartheta$ function by allowing a weight for each vertex. We provide a similar generalization of Duan, Severini, and Winter's $\tilde{\vartheta}$ on non-commutative graphs.…
We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of…
In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $\Gamma $. Furthermore, we give another proof of Dedekind Theorem for the group…
We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree…
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results…
Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…
In [19] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $\mathbb{F}_1$) to a so-called "loose graph" (which is a generalization of a graph). Several properties of…