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In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648-662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.  相似文献
2.
Let GG be a connected regular graph. Denoted by t(G)t(G) and Kf(G)Kf(G) the total graph and Kirchhoff index of GG, respectively. This paper is to point out that Theorem 3.7 and Corollary 3.8 from “Kirchhoff index in line, subdivision and total graphs of a regular graph” [X. Gao, Y.F. Luo, W.W. Liu, Kirchhoff index in line, subdivision and total graphs of a regular graph, Discrete Appl. Math. 160(2012) 560–565] are incorrect, since the conclusion of a lemma is essentially wrong. Moreover, we first show the Laplacian characteristic polynomial of t(G)t(G), where GG is a regular graph. Consequently, by using Kf(G)Kf(G), we give an expression on Kf(t(G))Kf(t(G)) and a lower bound on Kf(t(G))Kf(t(G)) of a regular graph GG, which correct Theorem 3.7 and Corollary 3.8 in Gao et al. (2012)  [2].  相似文献
3.
We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.  相似文献
4.
Two graphs are said to be L-cospectral (respectively, Q-cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph G is said to be L?DS (respectively, Q?DS) if there does not exist other non-isomorphic graph H such that H and G are L-cospectral (respectively, Q-cospectral). Let d1(G)d2(G)?dn(G) be the degree sequence of a graph G with n vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with d1(G){4,5}), if H is L-cospectral (respectively, Q-cospectral) with a connected graph G and d2(G)=2, then H has the same degree sequence as G. A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both L?DS, which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014).  相似文献
5.
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.  相似文献
6.
In this paper,we study the bases and base sets of primitive symmetric loop-free (generalized)signed digraphs on n vertices.We obtain sharp upper bounds of the bases,and show that the base sets of the classes of such digraphs are{2,3,...,2n-1}.We also give a new proof of an important result obtained by Cheng and Liu.  相似文献
7.
Let G be a graph. The Hosoya index Z(G) of a graph G is defined to be the total number of its matchings. In this paper, we characterize the graph with the smallest Hosoya index of bicyclic graphs with given pendent vertices. Finally, we present a new proof about the smallest Hosoya index of bicyclic graphs.  相似文献
8.
The Balaban index of a connected graph G is defined as J(G) =|E(G)|μ + 1∑e=uv∈E(G)1√DG(u)DG(v),and the Sum-Balaban index is defined as SJ(G) =|E(G)|μ + 1∑e=uv∈E(G)1√DG(u)+DG(v),where DG(u) =∑w∈V(G)dG(u, w), and μ is the cyclomatic number of G. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on n vertices are characterized, respectively.  相似文献
9.
Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 1215-1227], authors conjectured that D is the underlying digraph of S with exp(D) = 3 if and only if D is isomorphic to ED n,3,3 , where ED n,3,3 = (V, A) is a digraph with V = {1, 2, . . . , n}, A = {(1, i), (i, 1) | 3≤i≤n} ∪ {(2i-1, 2i), (2i, 2i-1) | 2≤i≤ n/2 } ∪ {(2, 3), (3, 2), (2, 4), (4, 2)}). In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.  相似文献
10.
Recently, the primitive symmetric signed digraphs on n vertices with the maximum base 2n and the primitive symmetric loop-free signed digraphs on n vertices with the maximum base 2n-1 are characterized, respectively. In this paper, the primitive symmetric signed digraphs with loops on n vertices with the base 2n-1 are characterized, and then the primitive symmetric signed digraphs on n vertices with the second maximum base 2n-1 are characterized.  相似文献
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