Comparative analyses of graph structured datasets underly diverse problems. Examples of these problems include identification of conserved functional components (biochemical interactions) across species, structural similarity of large biomolecules, and recurring patterns of interactions in social networks. A large class of such analyses methods quantify the topological similarity of nodes acros...

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investiga...

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radius.

6 The convolution layer 11 6.1 What is convolution? . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 Why to convolve? . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3 Convolution as matrix product . . . . . . . . . . . . . . . . . . . 15 6.4 The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . 17 6.5 Backward propagation: update the parameters . . . . . . . . ...

In this paper we give a combinatorial interpretation for the coefficient of sν in the Kronecker product s(n−p,p) ∗ sλ, where λ = (λ1, . . . , λ`(λ)) ` n, if `(λ) ≥ 2p − 1 or λ1 ≥ 2p − 1; that is, if λ is not a partition inside the 2(p − 1) × 2(p − 1) square. For λ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinat...

In this paper we give a combinatorial interpretation for the coefficient of sν in the Kronecker product s(n−p,p) ∗ sλ, where λ = (λ1, . . . , λl(λ)) ⊢ n, if l(λ) ≥ 2p − 1 or λ1 ≥ 2p − 1; that is, if λ is not a partition inside the 2(p − 1) × 2(p − 1) square. For λ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinat...

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radius.