اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدZeta functions of edge-free quotients of graphs
340
0
0.0
(
0
)
تأليف
Dmitry Zakharov
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the Ihara zeta function $zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $Xto X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $zeta(u,X)$ and that $zeta(u,X//G)$ divides $zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.
قيم البحث
اقرأ أيضاً
We define a zeta function of a finite graph derived from time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph.
We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (infinite) periodic graph.
Given a directed graph, an equivalence relation on the graph vertex set is said to be balanced if, for every two vertices in the same equivalence class, the number of directed edges from vertices of each equivalence class directed to each of the two
vertices is the same. In this paper we describe the quotient and lift graphs of symmetric directed graphs associated with balanced equivalence relations on the associated vertex sets. In particular, we characterize the quotients and lifts which are also symmetric. We end with an application of these results to gradient and Hamiltonian coupled cell systems, in the context of the coupled cell network formalism of Golubitsky, Stewart and Torok(Patterns of synchrony in coupled cell networks with multiple arrows. {SIAM Journal of Applied Dynamical Systems, 4 (1) (2005) 78-100).
Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated Cayley grap
h. Analogously, but much earlier, Dillon showed that f is bent if and only if the level curves of f had certain combinatorial properties (again, only when p is 2). The attempt is to investigate an analogous theory when p is greater than 2 using the (apparently new) combinatorial concept of a weighted partial difference set. More precisely, we try to investigate which properties of the Cayley graph of f can be characterized in terms of function-theoretic properties of f, and which function-theoretic properties of f correspond to combinatorial properties of the set of level curves, i.e., the inverse map of f. While the natural generalizations of the Bernasconi correspondence and Dillon correspondence are not true in general, using extensive computations, we are able to determine a classification in some small cases. Our main conjecture is Conjecture 67.
Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| geq (frac{k}{2} - frac{1}{k-1})|V(G)| - frac{k(k-3)}{2(k-1)}$. The class of graph
s for which this bound is tight, $k$-Ore graphs, contain a notably large number of $K_{k-2}$-subgraphs. Subsequent work attempted to determine the asymptotic density for $k$-critical graphs that do emph{not} contain large cliques as subgraphs, but only partial progress has been made on this problem. The second author showed that if $G$ is 5-critical and has no $K_3$-subgraphs, then for $varepsilon = 1/84$, $|E(G)| geq (frac{9}{4} + varepsilon)|V(G)| - frac{5}{4}$. It has also been shown that for all $k geq 33$, there exists $varepsilon_k > 0$ such that $k$-critical graphs with no $K_{k-2}$-subgraphs satisfy $|E(G)| geq (frac{k}{2} - frac{1}{k-1} + varepsilon_k)|V(G)| - frac{k(k-3)}{2(k-1)}$. In this work, we develop general structural results that are applicable to resolving the remaining difficult cases $6 leq k leq 32$. We apply our results to carefully analyze the structure of 6-critical graphs and use a discharging argument to show that for $varepsilon_6 = 1/1050$, 6-critical graphs with no $K_4$ subgraph satisfy $|E(G)| geq ( frac{k}{2} - frac{1}{k-1} + varepsilon_6 ) |V(G)| - frac{k(k-3)}{2(k-1)}$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
