Transcription of Chapter 17 Graphs and Graph Laplacians
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Chapter 17 Graphs and Graph Laplacians
Chapter 17 Graphs and Graph Directed Graphs , Undirected Graphs , IncidenceMatrices, Adjacency Matrices, Weighted GraphsDefinition graphis a pairG=(V, E),whereV={v1,..,vm}is a set ofnodesorvertices,andE V Vis a set of ordered pairs of distinct nodes (thatis, pairs (u, v)2V Vwithu6=v), edgee=(u, v), we lets(e)=ube thesourceofeandt(e)=vbe :Since an edge is a pair (u, v)withu6=v,self-loops are not , there is at most one edge from a nodeuto a 17. Graphs AND Graph LAPLACIANS1v4v5v1v2v3e1e7e2e3e4e5e6 Figure : every nodev2V,thedegreed(v)ofvis the numberof edges leaving or enteringv:d(v)=|{u2V|(v, u)2 Eor (u, v)2E}|.We abbreviated(vi) matrixD(G), isthe diagonal matrixD(G)=diag(d1,..,dm).For example, for confusion arises, we writeDinstead ofD(G). DIRECTED Graphs , UNDIRECTED Graphs , WEIGHTED GRAPHS735 Definition a directed graphG=(V, E),for any two nodesu, v2V,apath fromutovis asequence of nodes (v0,v1.)
graph with m vertices, n edges, and c connected com-ponents. For any orientation of G, if B is the in-cidence matrix of the oriented graph G, then c = dim(Ker(B>)), and B has rank m c. Furthermore, the nullspace of B> has a basis consisting of indica-
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