A natural question is therefore 'how far the Laplian spectrum of G can be described using the Laplian spectra of F and Hv.'We prove some results on this and show that the complete Laplian spectrum of G is also in some specific cases can be described. We conduct a study of both the ordinary spectrum (eigenvalues of the adjacency matrix) and the Laplian spectrum of coronas.
General introduction
He further discussed the case where one of the modified eigenvalues is algebraic connectivity. We investigate the Laplacian spectra of the product graphs obtained with the four products of the graphs mentioned above.
Preliminaries : the characteristic set
Fan [21] introduced and studied spectral integral variation in general graphs by adding an edge or a loop. If the spectral integral variation of Go occurs at one place by adding an edge e, then.
Spectral integral variation of graphs with a(G) increasing by 2
We see that spectral integral variation occurs in one place by adding the edge {n−1, n} to G, where the changed eigenvalue is a(G). So in G, spectral integral variation arises at one place by adding the edge {i, j}, where the changed eigenvalue is the algebraic connection.
Spectral Integral Variation in one place with a(G) unchanged
The following result describes a way to construct a graphG in which spectral integral variation occurs at one place by adding an edge where the modified eigenvalue is the algebraic connectivity. Proof: Suppose, on the other hand, that spectral integral variation of G occurs at one location by adding an edge where the changed eigenvalue is µ(G).
Trees with third smallest Laplacian eigenvalue 1
The following result places a bound on the diameter of a tree with 1 as the third smallest Laplian eigenvalue. The following result characterizes all trees of diameter 4, with 1 as the third smallest Laplian eigenvalue. Any tree T of diameter 4 that is not of the form described in the theorem is either isomorphic to ¯T or can be obtained from ¯T by adding hanging vertices.
Our next result characterizes all trees with diameter 5 and with 1 as the third smallest Laplacian eigenvalue. Any tree T of diameter 5 that is not of the form described in the declaration is either isomorphic to ˆT or can be obtained from ˆT by adding adjacent vertices.
Spectral integral variation of trees with λ 3 increasing by 2
Then the spectral integral change of T occurs at a place by adding an intermediate edge where the changed eigenvalue is the third smallest Laplacian eigenvalue if and only if T is one of T2,l,k, T3,l,k, or T4,l ,k and i, j are two pendant vertices adjacent to the same vertex. Thus, using Theorem 2.1.3, the spectral integral variation of T occurs at a location by adding an edge between i and j where the Laplacian eigenvalue 1 changes to 3. Conversely, let the spectral integral variation of T occur at a location by adding an edge between i and j where the shifted eigenvalue is λ3.
Then spectral integral variation of T occurs at one location by adding an edge intermediateandj where the modified eigenvalue is the third smallest Laplacian eigenvalue if and only if T is a tree obtained from a path of length two or three by adding two adjacent vertices i and j to one end (that is, T is either T2,1,2 or T3,1,2 andi, j are two adjacent vertices adjacent to the same vertex). The Laplacian spectra of graphs obtained by operations such as complement, disjoint union, and join are well studied.
Graphs with pockets
Let G[r, l, Hv] be the graph obtained by attaching a copy of Hv to each except the central vertex of Sr,l, each at vertex v of Hv. Let G=G[r, l, Hv] be the graph obtained by attaching a copy of Hv to each but the central vertex of Sr,l, each at the vertex of Hv. Let G = G[r, Hv] be the graph obtained by attaching a copy of Hv to each vertex of degree 1 of Sr+1, each at vertex v of Hv.
G[F, u1, Hv] is the graph obtained by attaching a copy of Hv to the vertex of F atv. G[F, u1,S5] is the graph obtained by attaching a copy of S5 to the same vertex of F in the central vertex of star S5.
Some applications
This implies that µ(G) is one of the then+keigen values of L(G) that are independent of the graph Hv. Define Fk[H] as a graph on m+n vertices obtained by connecting some k,1 ≤ k ≤ m vertices of F to each node of H. Let G=Fk[H] be the graph obtained by to join some k,1≤k≤m vertices from F to every vertex from H.
Thus µ(G) is either n+k or one of the m+ 1 eigenvalues of L(G) that are independent of the graph H. Like the graph operations, graph products are also used to construct many important classes of graphs.
Graph cartesian product
The next result, which follows from Theorem 4.2.2, describes the characteristic set of the Cartesian product. It also follows from theorem 4.2.2 that if X and Y are Fiedler vectors of F and H, respectively, then 11⊗Y and X⊗11 are Fiedler vectors of F¤H. As a direct application of Theorem 4.2.3 we have the following result describing the case of the simple algebraic relation of the Cartesian product.
In the former case, the characteristic set of F¤H is the Cartesian product of the characteristic set of F with the graph H. Furthermore, we have a description of the Fiedler vector and the characteristic set of this graph (see Corollary 4.2.5).
Graph categorical product
The following result describes the Laplacian matrix of F ×H in terms of the Laplacian matrices of F and H. Proposition 4.3.5 Let F be a regular graph of m vertices with regularity r and H be any graph of n vertices. In general, the categorical product of two Laplacian integral graphs is not necessarily Laplacian integral.
The following result shows that Kn×P3, n≥ 3, is not Laplacian integral even though Kn and P3 are Laplacian integral graphs. The following well-known result [25] describes the Laplacian spectrum of a cycle and was used later.
Graph strong product
Conversely, if H is regular, then again by Anderson and Morley's result µ(H) = 2∆(H) and thus equality in (4.3) holds. In general, the complete description of the Laplacian spectrum of F£H cannot be obtained from the Laplacian spectra of F and H alone. The following result describes the complete Laplacian spectrum of F£H using the Laplacian spectra of F and H in the case of F and H Regular race.
But in general, the strong product of two Laplace integral graphs does not produce a Laplace integral graph. With a similar argument as in Theorem 4.3.7 it can be proven that the graphP3£P3 is not a Laplace integral.
Graph lexicographic product
From the definition of lexicographic product it follows that: Let F and H be two non-trivial graphs with at least two vertices. Our next result describes the complete Laplacian spectrum of F[H] using the Laplacian spectra of F and H. The corona F ◦H of F and H is defined as the graph obtained by taking one copy of F and n copies of H, and then connect the ith vertex of F to every vertex of the ith copy of H.
Note that the corona operation is a specific case of the operation defined in definition 3.2.1 of chapter 3. In section 5.3 we prove some structural results on the Fiedler vectors of the coronas and offer an application.
Next, we construct infinitely many pairs of non-isomorphic graphs with the same spectrum and the same Laplacian spectrum using the corona operation. The following result describes the Laplacian eigenvalues and Laplacian eigenvectors of F◦H using the Laplacian eigenvalues and Laplacian eigenvectors offF and H .
Some applications
Construction of Type I graphs with nonisomorphic Perron branches
In view of this result, all we need is to take a tree of type I T at more than 2 vertices with non-isomorphic Perron branches and any graph H. In this way we can construct an infinite class of graphs of type I with non-isomorphic Perron branch. In particular, considering the tree T in Figure 5.4, which is known to be Type I with non-isomorphic Perron branches, and taking H as an isolated vertex, we see that T, T ◦ H,(T◦H)◦ H,.
Cospectral and Laplacian cospectral graphs
There are also examples of pairs of graphs which are both cospectral and Laplacian cospectral (See van Dam and Haemers [66]). Note that if two graphs are regular and cospectral, then they are also Laplacian cospectral. Let F and H be two ionisomorphic cospectral and Laplacian cospectral graphs (take for example the two graphs given in Figure 5.6).
So for each i the graphs Fi, Hi are non-isomorphic cospectral and Laplacian cospectral irregular graphs. If we replace B with any other graph, we can get infinite pairs of non-isomorphic irregular graphs without any adjacent vertex which are cospectral and Laplacian cospectral.
Trees with property (SR)
We provide suitable examples that prove that a graph with property (R) is not necessarily the crown of two graphs and is not necessarily bipartite. A close inspection of P4 leads us to the following result, which gives a class of bipartite graphs with property (R). It turns out that every property tree (SR) is of the form T ◦K1 for some tree T.
Since P(G;x) is monic and the leading coefficient ofxnP(G;x1) is ±1, it follows that P(G;x) =±xnP(G;x1) and the conclusion follows. Thus, the number of pairwise edge-connected subsets of size k−1 or T exceeds 2k−1, which is a contradiction and the claim is justified.
Bipartite graphs with a unique perfect matching
In Section 6.3, we give a combinatorial description of the inverse of the adjacency matrix of a bipartite graph with a unique perfect matching. We will give an answer to this that requires describing the inverse of the adjacency matrix. A combinatorial description of the inverse of the adjacency matrix of a nonsingular tree is given in Buckley, Doty and Harary [14, Theorem 3] and in Pavlikova [62, Theorem 1].
The following result gives a description of the inverse of the adjacency matrix of a bipartite connected graph with a unique perfect matching. Hence, we have the following result which gives the combinatorial description of the inverse of a non-singular tree.
Trees with property(R)
Pati, On Algebraic Connection and Spectral Integral Variations of Graphs, Linear Algebra and Its Applications. Pati, On Algebraically Connected Graphs Corresponding to Minimum Edge Density, Linear Algebra and Its Applications. Guo, The effect of a graph's Laplacian spectral radius by adding or grafting edges, Linear Algebra and its Applications.
Shao, On the spectral radius of fixed-diameter trees, Linear Algebra and Its Applications, article in press, (2005). Li, Two-place spectral rational variation for an adjacency matrix is not possible, Linear algebra and its applications.
