By Toshiaki Adachi, Hideya Hashimoto, Milen J Hristov

This quantity includes contributions by means of the most contributors of the 4th overseas Colloquium on Differential Geometry and its similar Fields (ICDG2014). those articles conceal contemporary advancements and are committed as a rule to the research of a few geometric constructions on manifolds and graphs. Readers will discover a vast evaluate of differential geometry and its dating to different fields in arithmetic and physics.

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**Additional resources for Current Developments in Differential Geometry and its Related Fields: Proceedings of the 4th International Colloquium on Differential Geometry and its Related Fields**

**Example text**

Under the assumption in the assertion (1), we have D(p) = D(a) . If we (1,1) (1,1) denote by AG∗ the matrix representation of AG∗ , we get (1,1) (1,1) AG∗ = A(G∗ )(p) D(p)A(G∗ )(a) = tAG (1,1) (1,1) = ΦAG(p) D(a)AG(a) Φ−1 = ΦAG Φ−1 . (1,1) We hence obtain AG∗ = ϕ∗ ◦ AG ◦ (ϕ∗ )−1 . (p) Under the assumption in the assertion (2), we have D(p) = dG I and (1,1) (1,1) (a) D(a) = dG I. Hence we obtain QG∗ = ϕ∗ ◦ QG ◦ (ϕ∗ )−1 . Therefore, we get the conclusions. 1. When G is regular and is isomorphic to its dual G∗ , our (p,q) (p,q) proof shows that ∆QG ◦ ϕ∗ = ϕ∗ ◦ ∆QG∗ .

4. Duality of adjacency and transition operators For a finite K¨ ahler graph G = (V, E (p) ∪ E (a) ), we denote by Pp,q (v; G) the set of all (p, q)-primitive bicolored paths whose origin is v ∈ V . We (p,q) define the adjacency operator AG : C(V ) → C(V ) and the probabilistic (p,q) transition operator QG : C(V ) → C(V ) acting on the set C(V ) of all complex-valued functions by (p,q) AG ωG (γ)f t(γ) , f (v) = γ∈Pp,q (v;G) page 28 August 27, 2015 9:16 Book Code: 9748 – Current Developments in Differential Geometry ws-procs9x6˙ICDG2014 ¨ LAPLACIANS FOR FINITE REGULAR KAHLER GRAPHS (p,q) QG 1 f (v) = ωG (γ) 29 ωG (γ)f t(γ) .

Rio, H. Hashimoto, T. Koda & T. , 30–40, (World Scientific, Singapore, 2005). 3. , A discrete model for K¨ahler magnetic fields on a complex hyperbolic space, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, K. S. Gerdjikov & S. Dimiev eds, 1–9 (World Scientific, Singapore, 2009). , A theorem of Hadamard-Cartan type for K¨ahler magnetic 4. fields, J. Math. Soc. Japan 64, 969–984 (2012). 5. T. , Zeta functions for a K¨ahler graph, preprint (2013). page 43 September 14, 2015 14:11 Book Code: 9748 – Current Developments in Differential Geometry ws-procs9x6˙ICDG2014 44 T.