A mixed graph G^(-) is obtained by orienting some edges of G, where G is the underlying graph of G^(-) . The positive inertia index, denoted by p~+( G), and the negative inertia index, denoted by n~-(G^(-) ), of a mix...A mixed graph G^(-) is obtained by orienting some edges of G, where G is the underlying graph of G^(-) . The positive inertia index, denoted by p~+( G), and the negative inertia index, denoted by n~-(G^(-) ), of a mixed graph G^(-) are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of G^(-) , respectively. In this paper, the positive and negative inertia indices of the mixed unicyclic graphs are studied. Moreover, the upper and lower bounds of the positive and negative inertia indices of the mixed graphs are investigated, and the mixed graphs which attain the upper and lower bounds are characterized respectively.展开更多
A fractional matching of a graph G is a function f: E(G)→[0,1] such that for each vertex v, ∑eϵΓG(v)f(e)≤1.. The fractional matching number of G is the maximum value of ∑e∈E(G)f(e) over all fractional matchings ...A fractional matching of a graph G is a function f: E(G)→[0,1] such that for each vertex v, ∑eϵΓG(v)f(e)≤1.. The fractional matching number of G is the maximum value of ∑e∈E(G)f(e) over all fractional matchings f. Tian et al. (Linear Algebra Appl 506:579–587, 2016) determined the extremal graphs with minimum distance Laplacian spectral radius among n-vertex graphs with given matching number. However, a natural problem is left open: among all n-vertex graphs with given fractional matching number, how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this paper, we solve these problems completely.展开更多
In [6],Guo and Tan have shown that 2 is a Laplacian eigenvalue of any tree with perfect matchings.For trees without perfect matchings,we study whether 2 is one of its Laplacian eigenvalues.If the matchingnumber is 1 o...In [6],Guo and Tan have shown that 2 is a Laplacian eigenvalue of any tree with perfect matchings.For trees without perfect matchings,we study whether 2 is one of its Laplacian eigenvalues.If the matchingnumber is 1 or 2,the answer is negative;otherwise,there exists a tree with that matching number which has (hasnot) the eigenvalue 2.In particular,we determine all trees with matching number 3 which has the eigenvalue2.展开更多
The （conditional） matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph （with no isolated vertices） that has neither perfect matchings nor almost perfect matc...The （conditional） matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph （with no isolated vertices） that has neither perfect matchings nor almost perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented k-ary n-cubes with even k ≥ 4.展开更多
This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph th...This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result （Theorem 13）, Grone and Merris＇ conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221-229 （1994）] is partially proved.展开更多
基金the National Natural Science Foundation of China (Nos. 11971054 and 12161141005)the Fundamental Research Funds for the Central Universities (No. 2016JBM071)。
文摘A mixed graph G^(-) is obtained by orienting some edges of G, where G is the underlying graph of G^(-) . The positive inertia index, denoted by p~+( G), and the negative inertia index, denoted by n~-(G^(-) ), of a mixed graph G^(-) are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of G^(-) , respectively. In this paper, the positive and negative inertia indices of the mixed unicyclic graphs are studied. Moreover, the upper and lower bounds of the positive and negative inertia indices of the mixed graphs are investigated, and the mixed graphs which attain the upper and lower bounds are characterized respectively.
基金This work is supported by the Science and Technology Program of Guangzhou,China(No.202002030183)the Guangdong Province Natural Science Foundation(No.2021A1515012045)the Qinghai Province Natural Science Foundation(No.2020-ZJ-924).
文摘A fractional matching of a graph G is a function f: E(G)→[0,1] such that for each vertex v, ∑eϵΓG(v)f(e)≤1.. The fractional matching number of G is the maximum value of ∑e∈E(G)f(e) over all fractional matchings f. Tian et al. (Linear Algebra Appl 506:579–587, 2016) determined the extremal graphs with minimum distance Laplacian spectral radius among n-vertex graphs with given matching number. However, a natural problem is left open: among all n-vertex graphs with given fractional matching number, how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this paper, we solve these problems completely.
基金The project item of scientific research fund for young teachers of colleges and universities of Anhui province (Grant No.2003jq101) and the project item of Anhui University fund for talents group construction,and National Natural Science Foundation of Ch
文摘In [6],Guo and Tan have shown that 2 is a Laplacian eigenvalue of any tree with perfect matchings.For trees without perfect matchings,we study whether 2 is one of its Laplacian eigenvalues.If the matchingnumber is 1 or 2,the answer is negative;otherwise,there exists a tree with that matching number which has (hasnot) the eigenvalue 2.In particular,we determine all trees with matching number 3 which has the eigenvalue2.
文摘The （conditional） matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph （with no isolated vertices） that has neither perfect matchings nor almost perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented k-ary n-cubes with even k ≥ 4.
基金Supported by National Natural Science Foundation of China （Grant No. 10871204） and the Fundamental Research Funds for the Central Universities （Grant No. 09CX04003A）
文摘This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result （Theorem 13）, Grone and Merris＇ conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221-229 （1994）] is partially proved.
