Connected Q-integral graphs with maximum edge-degree less than or equal to 8 |
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Institution: | Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh-221005, India |
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Abstract: | Graphs with integral signless Laplacian spectrum are called Q-integral graphs. The number of adjacent edges to an edge is defined as the edge-degree of that edge. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In 2019, Park and Sano 16] studied connected Q-integral graphs with the maximum edge-degree at most six. In this article, we extend their result and study the connected Q-integral graphs with maximum edge-degree less than or equal to eight. Further, we give an upper bound and a lower bound for the maximum edge-degree of a connected Q-integral graph with respect to its Q-spectral radius. As a corollary, we show that the Q-spectral radius of the connected edge-non-regular Q-integral graph with maximum edge-degree five is six, which we anticipate to be a key for solving the unsolved problem of characterizing such graphs. |
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Keywords: | Edge-degree Integral graph Signless Laplacian matrix |
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