19,945 research outputs found
An entropy based proof of the Moore bound for irregular graphs
We provide proofs of the following theorems by considering the entropy of
random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple
graph with n vertices, girth g, minimum degree at least 2 and average degree d:
Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If
g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G =
(V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R =
|V_R|, minimum degree at least 2 and the left and right average degrees d_L and
d_R. Then, n_L \geq \Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{i/2} n_R \geq
\Sum_{i=0}^{r-1}(d_L-1)^{i/2}(d_R-1)^{i/2}Comment: 6 page
BPS Graphs: From Spectral Networks to BPS Quivers
We define "BPS graphs" on punctured Riemann surfaces associated with
theories of class . BPS graphs provide a bridge between
two powerful frameworks for studying the spectrum of BPS states: spectral
networks and BPS quivers. They arise from degenerate spectral networks at
maximal intersections of walls of marginal stability on the Coulomb branch.
While the BPS spectrum is ill-defined at such intersections, a BPS graph
captures a useful basis of elementary BPS states. The topology of a BPS graph
encodes a BPS quiver, even for higher-rank theories and for theories with
certain partial punctures. BPS graphs lead to a geometric realization of the
combinatorics of Fock-Goncharov -triangulations and generalize them in
several ways.Comment: 48 pages, 44 figure
Analysis Of The Girth For Regular Bi-partite Graphs With Degree 3
The goal of this paper is to derive the detailed description of the
Enumeration Based Search Algorithm from the high level description provided in
[16], analyze the experimental results from our implementation of the
Enumeration Based Search Algorithm for finding a regular bi-partite graph of
degree 3, and compare it with known results from the available literature. We
show that the values of m for a given girth g for (m, 3) BTUs are within the
known mathematical bounds for regular bi-partitite graphs from the available
literature
A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter
AMS classifications: 05C50, 05E99;graphs;spectral radius;diameter;bound;degree/diameter
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