图的距离(无符号)拉普拉斯谱

We will use algebraic,combinatorial and graph theory and apply results attained in matrix,spectrum of graph and algebraic graph theory to study distance laplacian and distance signless laplacian. Mainly they contain the following five aspects:①Determine the graphs with minimum distance Laplacian spectral radius among the trees, unicyclic graphs and bicylic graphs, respectively;②Characterize the graphs with minimum distance (signless) laplacian spectral radius among graphs whose structures are special such as bipartite graphs,graphs with fixed connectivity and graphs without pendant vertex;③Study the upper and lower bound of distance (signless) laplacian spectral radius of graphs to attain their expression with such the structural parameters as diameter,number of color etc;④Study the graphs whose some eigenvalue of distance (signless) laplacian matrix is limited;⑤Study distance (signless) laplacian spectra of the graphs obtained by operation such as the product graphs etc. Our research objectives are to obtain distance (signless) laplacian spectral characterization of graphs and characterize the graphs with the extremal distance (signless) laplacian spectral radius. The distance (signless) laplacian of graphs have important applied value in such research field as applied mathematics,physics and chemistry. On the other hand, it is also essential to perfect the spectral theory of graphs. Up to now research of distance (signless) laplacian of graphs have not been entirety spread, and so this program will promote the researches in the field.

本项目拟采用代数、组合及图论的方法，利用矩阵论、图谱理论与代数图论的相关结果，研究图的距离拉普拉斯谱和距离无符号拉普拉斯谱。主要研究以下五个方面的内容：①确定树、单圈图及双圈图的距离拉普拉斯谱半径取得极值时的极图；②刻画二部图、给定连通度、无悬挂点等结构特殊的图的距离（无符号）拉普拉斯谱半径达到极值时的极图；③对图的距离（无符号）拉普拉斯谱半径的上下界展开研究，尤其是建立它与图的结构参数，如直径、色数等的关系；④对某个特征值限于一定范围内的图类进行研究；⑤对各种由图运算得到的图，如积图的距离（无符号）拉普拉斯图谱展开研究。以期完成图的距离（无符号）拉普拉斯谱特征的刻画和距离（无符号）拉普拉斯谱图特征的刻画。 图的距离拉普拉斯谱在应用数学、物理及化学等方面都具有很大的使用价值，图谱理论也需要完善该领域的研究结果。目前，图的距离拉普拉斯谱的研究还未真正展开，本项目必将推动该领域的研究工作

本项目采用代数、组合及图论的方法，利用矩阵论、图谱理论与代数图论的相关结果，研究图的距离拉普拉斯谱和距离无符号拉普拉斯谱。主要得到以下十个方面的研究结果：①刻画了给定连通度和匹配数的二部图的距离拉普拉斯谱半径达到极大值时的极图；②确定了带有割边的图的距离拉普拉斯谱半径达到极大值时的极图；③分别描述了给定连通度和最小度的图的距离无符号拉普拉斯谱半径达到极小值时的极图；④得到了给定连通度和双色数的有向图的距离无符号拉普拉斯谱半径达到极小值时的极图；⑤给出了图和有向图的距离无符号拉普拉斯谱半径的上下界；⑥分别刻画了给定独立数，连通度和匹配数的二部图的（无符号）拉普拉斯谱半径达到极大值时的极图；⑦分别刻画了给定连通度，块数和悬挂点数的图的无符号拉普拉斯谱半径达到极大值时的极图；⑧刻画了带有给定悬挂点数的仙人掌图的（无符号）拉普拉斯谱半径达到极大值时的极图；⑨分别算出了萤火虫图的距离矩阵及距离（无符号）拉普拉斯矩阵的两个最大的特征值的和的下界；⑩刻画了T型树的线图的同谱图。. 上述研究结果分别发表在SCI或国内核心期刊上，预期的研究目标已经完成，符合结题的要求。