Doubly stochastic graph matrices,II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))?¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

Intrinsic ordering,combinatorial numbers and reliability engineering

A new algorithm for evaluating the top event probability of large fault trees (FTs) is presented. This algorithm does not require any previous qualitative analysis of the FT. Indeed, its efficiency is independent of the FT logic, and it only depends on the number n of basic system components and on their failure probabilities. Our method provides exact lower and upper bounds on the top event probability by using new properties of the intrinsic order relation between binary strings. The intrinsic order enables one to select binary n  -tuples with large occurrence probabilities without necessity to evaluate them. This drastically reduces the complexity of the problem from exponential (2ⁿ$2^n$ binary n-tuples) to linear (n Boolean variables). Our algorithm is mainly based on a recursive formula for rapidly computing the sum of the occurrence probabilities of all binary n-tuples with weight m whose 1s are placed among the k right-most positions. This formula, as well as the balance between accuracy and computational cost, is closely related to the famous Pascal’s triangle.     

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

Distance unimodular equivalence of graphs

Let G be a connected graph and D(G) be its distance matrix. In this article, the Smith normal forms of the integer matrices D(G) are determined for trees, wheels, cycles, complements of cycles and are reduced for complete multipartite graphs.     

On the spectral radius of tricyclic graphs with a fixed diameter

A tricyclic graph is a connected graph with n vertices and n + 2 edges. In this article, we determine graphs with the largest spectral radius among the n-vertex tricyclic graphs with given diameter d.