TRIANGLESSTOCHASTIC matricesBIPARTITE graphsLOGICAL predictionEDGES (Geometry)Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr 1-free graph on at least r 1 vertices and m edges, then λ12(G) λ22(G) ≤ (r 1)/r 2m, ...
Since α(G)=2, G¯ has no triangles. It is well-known that the complement of an odd cycle Cn with n≥5 has at least three positive eigenvalues. Therefore, by interlacing, G¯ has no induced odd cycle, so G¯ is bipartite and the adjacency matrix A of G has the following ...
The coefficients of the characteristic polynomial P(G;x) of a connected graph G satisfy: c1=0,−c2 is the number of edges and −c3 is twice the number of triangles of G. Let A be the adjacency matrix of a graph G. It is well known that the (i,j)-th element aij(k) of the...
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We examine the general dimensional relation between the shape space of the n-simplexes and the subspace of their achiral sets. It is found that the only simplexes that can be partitioned into ''left-handed'' and ''right-handed'' classes are the triangles in Euclidean 2-space....
In this paper, we first present a new upper bound for λ 1 ( G ) when each edge of G belongs to at least t ( t ≥ 1 ) triangles. Some new upper and lower bounds on q 1 ( G ), q 1 ( G ) + q 1 ( G c ) are determined, respectively. We also compare our results in ...
{m,n}\\) is the unique graph which has maximal value of \\(S_{2}(G)\\), where \\(G_{m,n}\\) is a graph of order n with m edges which has \\(m-n+1\\) triangles with a common edge and \\(2n-m-3\\) pendent edges incident with one end vertex of the common edge...
In this paper, we first present a new upper bound for ~(λ1)(G) when each edge of G belongs to at least t(t<1) triangles. Some new upper and lower bounds on ~(q1)(G), ~(q1)(G)+~(q1)(Gc) are determined, respectively. We also compare our results in this paper with some ...
This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen....
At last, we characterize the unique graph having the maximum spectral radius among the set of n n -vertex non-bipartite graphs with odd girth at least 2k+3, 2k+3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. ...