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In this paper, we characterize all similarity relations when m≥3, obtain some interesting properties which are different from the matrix case (m=2), and show that some of the well-known results of matrices cannot be extended to tensors of order m≥3 . [ABSTRACT FROM AUTHOR]

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag (D) be the diagonal matrix with outdegrees of the vertices of D and Q(D)=diag (D)+A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give a sharp bound on q(D) where D has a given outdegree sequence and compare the bound with known bounds. We establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the extremal graph. In addition, we also determine the unique digraph which achieves the minimum (or maximum), the second minimum (or maximum), the third minimum, the fourth minimum spectral radius and signless Laplacian spectral radius among all strongly connected digraphs, and answer the open problem proposed by Lin and Shu [14] [ABSTRACT FROM AUTHOR]

In 2012, Abdo and Dimitrov defined the total irregularity of a graph G = (V, E) as irrt (G) = (1/2)Σ u, VϵV dG(U) - dG (V), where d G (U) denotes the vertex degree of a vertex u ϵ V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs on n vertices. [ABSTRACT FROM AUTHOR]

Thenullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph Γ(∞(p, q, l)), obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs. [ABSTRACT FROM AUTHOR]

Presents a study which proposed a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations. Significance of dynamic surface modeling in geometric modeling; Focus of traditional geometric modeling methods; Description of a modeling method popular for parametric surfaces.