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**d=(j,i)**t(d)=j o(d)=i Compare with :**3. Spectral function**Note : 1. 2. The poles of det (I2B - U()) coincide with the poles of detU 3. The zeros of det (I2B - U()) coincide with the zeros of the characteristic polynomial det ( IV - L) 4. The rhs of is a real and bounded function for real . 5. The last step is to expand det (I2B - U()) in periodic orbits Comment: If you are worried because of convergence issues etc, add to a small negative Imaginary part : The set of primitive periodic orbits on G ap() : amplitude**Combining and + periodic orbit**product we get Comments: 1. All periodic orbits are included in the function. 2. Functional equation for v regular graphs: Define: and 3. Analogous to the function for quantum graphs 4. Connects to a corresponding ( dependent) “classical” dynamics in terms of the bi-stochastic matrix M Md’,d() = | Ud’,d() |2 as for quantum graphs.**Compare with:**See also: A. A. Terras & H.M. Stark , M. Kotani & T. Sunada AAT : forthcoming book. All periodic orbits, weighted by scattering amplitudes**3. Trace formula**Periodic orbits sum (p) : “action” (function of ) ap : “stability amplitude”(function of ) “Weyl” Explicit forms of the “actions” (p) and the “stability amplitudes” can be written down in terms and ap. They follow from the definition of the vertex scattering matrices (i)d’,d given above. Smooth+Fluctuations (2-periodic orbits) “Smooth” v (=40) regular graph : Why? See previous lectures for an answer 1-/v**A short list of open problems:**• Challenges for combinatorial graph theory. • 1. Counting statistics of nb t-periodic cycles. • 2. The Random Waves conjecture. • - Other models of random graphs – The G(V,p) Ensemble. • Localization, spectral measures. • Isospectral graphs and the nodal count: Can nodal counts • distinguish between isospectral graphs?**היום קצר, והמלאכה מרובה ....**לא עליך המלאכה לגמור, ולא אתה בן חורין להבטל ממנה. The day is short and the work is aplenty… it is not up to you to finish it, but you are not free to remain idle.