The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...

By the theorem of Mantel (1907) it is known that a graph with n vertices and ⌊n24⌋+1 edges must contain triangle. A Erdős gives strengthening: there are not only one, but at least ⌊n2⌋ triangles. We give further improvement: if no vertex contained by all triangles then n−2 them. There some natural generalizations when (a) complete graphs considered (rather than triangles), (b) has t extra (not ...

Three computer algorithms are presented. One reduces a network N to its interior, I. Another counts all the triangles in network, and last randomly generates networks similar given just interior However, these not usual numeric programs that manipulate matrix representation of network; they set-based. Union meet essential binary operators; contained_in is basic relational comparator. The I show...