Abstract: The nullity η(G) of a graph is the occurrence of zero as an eigenvalue in its spectra. A zero-sum weighting of a graph G is real valued function, say f from vertices of G to the set of real numbers, provided that for each vertex of G the summation of the weights f(w) over all neighborhood w of v is zero for each v in G.A high zero-sum weighting of G is one that uses maximum number of non-zero independent variables. If G is graph with an end vertex, and if H is an induced subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then, η(G)= η(H). In this paper, a high zero-sum weighting technique and the endvertex procedure are applied to evaluate the nullity of t-tupple and generalized t-tupple graphs are derived and determined for some special types of graphs,
Also, we introduce and prove some important results about the t-tupple coalescence, Cartesian and Kronecker products of nut graphs.
Abstract: In this paper, we represent protein structure by using
graph. A protein structure database will become a graph database.
Each graph is represented by a spectral vector. We use Jacobi
rotation algorithm to calculate the eigenvalues of the normalized
Laplacian representation of adjacency matrix of graph. To measure
the similarity between two graphs, we calculate the Euclidean
distance between two graph spectral vectors. To cluster the graphs,
we use M-tree with the Euclidean distance to cluster spectral vectors.
Besides, M-tree can be used for graph searching in graph database.
Our proposal method was tested with graph database of 100 graphs
representing 100 protein structures downloaded from Protein Data
Bank (PDB) and we compare the result with the SCOP hierarchical
structure.