Spectral Characterization of Some Cubic Graphs

Abstract

It is proved that the Cartesian product of an odd cycle with the complete graph on 2 vertices, is determined by the spectrum of the adjacency matrix. We also present some computational results on the spectral characterization of cubic graphs on at most 20 vertices.

Signless Laplacian spectral characterization of line graphs ofT-shape trees

“…The structure with I (G) ≤ 1 that remains to be considered is that of H -shaped trees, denoted here by H a,b,c,d,e . Moreover, the starlike trees T l 1 ,l 2 ,l 3 ,l 4 , the dumbbell graphs D l,g 1 ,g 2 and the theta graphs θ a,b,c have been recently investigated in [10,23,27,30,32].…”

“…Proof Note that the result can be obtained indirectly from the A-variant [27] by means of Lemma 3.2. Let ϕ(P n , κ) = p n .…”

Spectral characterizations of graphs with small spectral radius

“…(1) By Lemma 2.3 again we have φ(P n , λ) = λφ(P n−1 , λ) − φ(P n−2 , λ), which can be read as a recurrence equation with boundary conditions φ(P 1 ) = λ and φ(P 0 ) = 1, whose solution (see [17]) is…”

Note on the spectral characterization of some cubic graphs with maximum number of triangles

“…Generally speaking, not all regular graphs are DAS (e.g., [3, p. 157]) and distance regular graphs have attracted more attention, see [1,6] for examples. It is also proved that the Cartesian product of an odd cycle with the complete graph K 2 is DAS [12]. In this note, we investigate the structure of cubic graphs of given order that have maximum number of triangles, and characterize the extremal cubic graphs.…”