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Q-integral graphs with edge-degrees at most five
Abstract
We consider the problem of determining the Q-integral graphs, i.e. the graphs with integral signless Laplacian spectrum. We find all such graphs with maximum edge-degree 4, and obtain only partial results for the next natural case, with maximum edge-degree 5.
... The integral and L-integral graphs are well studied in the literature. On the other hand, the graphs with integral Q-spectrum are studied in exactly two papers [9] and [10], so far. Since the matrix Q is positive semidefinite, the Q-spectrum consists of non-negative values. ...
... In Section 3, we identify some infinite series of ALQ-integral graphs. All (2, s)-semiregular bipartite Q-integral graphs are determined in Section 4. Some possible Q-spectra of connected Q-integral (3, 4)-semiregular bipartite graphs obtained in [9] are considered in Section 5. In addition, we give the possible Q-spectra of connected Qintegral (3, 5)-semiregular bipartite graphs and consider some of them. ...
... Following [9] and [10], we list some results regarding Q-integral graphs. All Q-integral graphs with maximum edge-degree at most 4 are known; exactly 26 of them are connected. ...
We consider the problem of determining the Q-integral graphs, i.e., the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all (2,s)-semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results concerning (3,4) and (3,5)-semiregular bipartite graphs with the same property.
... , q n−1 , q n ), is the sequence of the eigenvalues of Q (G) displayed in non-increasing order q 1 · · · q n−1 q n . Results related to Q and its spectrum can be found in [1][2][3][4]. ...
... A graph G is Q -integral if the spectrum of Q (G) consists only of integers [2,3]. For a regular graph G, it is well known that G is integral if and only if it is Laplacian integral. ...
... From Corollary 2.1, we have that the graph MCS a b,n is Q -integral if and only if (a + 2(b − 1) − nb) 2 + 4abn is a perfect square, which happens if a, b and n satisfy any of the conditions given in the proposition. Fig. 3 displays the multiple complete split-like graph MCS 2 3,3 whose Q -spectrum is Sp Q (MCS 2 3,3 ) = (12, 9, 6 2 , 3 7 ). ...
In this article, we characterize all signless Laplacian integral graphs (here called QQ-integral graphs) in the following classes: complete split graphs, multiple complete split-like graphs, extended complete split-like graphs, multiple extended split-like graphs. All these graphs were defined by Hansen et al. [P. Hansen, H. Melot, D, Stevanović, Integral complete split graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 89-95]. Also, we characterize all QQ-integral graphs of the form Kn1∨(Kn2∪Kn3)Kn1∨(Kn2∪Kn3), where Kni,i=1,2,3, is a complete graph on nini vertices. These characterizations allow us to exhibit many infinite families of QQ-integral graphs.
... ≥ q n−1 ≥ q n . A graph G is Q-integral if the spectrum of Q(G) consists only of integers [2]. ...
... Example 1.1 The graphs G 1 and G 2 shown in the figure bellow reflect this situation. Their Q-spectra are, respectively, Sp(Q(G 1 )) = (4, 3 (2) , 1 (2) , 0) and Sp(Q(G 2 )) = (5, 3 (2) , 2, 1, 0) ...
... Example 1.1 The graphs G 1 and G 2 shown in the figure bellow reflect this situation. Their Q-spectra are, respectively, Sp(Q(G 1 )) = (4, 3 (2) , 1 (2) , 0) and Sp(Q(G 2 )) = (5, 3 (2) , 2, 1, 0) ...
Let G be a connected graph with two nonadjacent vertices and G′ be the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′, respectively. Hence, the sum of the eigenvalues of Q′ is the sum of the eigenvalues of Q plus 2. Since none of the eigenvalues of Q can decrease if an edge is added to G, it is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by 2, or when two Q-eigenvalues are increased by 1 one each. In this article we give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs.
... The integral graphs are well studied in the literature. On the other hand, the graphs with integral Q-spectrum are studied in [4], [5], [8], [9] and [10], so far. Here we extend those results. ...
... Following [8], [10] and [9], we list some results regarding Q-integral graphs. All Q-integral graphs with maximum edge-degree at most 4 are known; exactly 26 of them are connected. ...
... Finally, if G is an (r, s)-semiregular bipartite graph which contains q quadrangles and h hexagons then for the spectral moments T k (k = 4, 5, 6) we have (cf. [8], Lemma 3.2) ...
A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. We establish some general results regarding signless Laplacians of semiregular bipartite graphs. Especially, we consider those semiregular bipartite graphs with integral signless Laplacian spectrum. In some particular cases we determine the possible Q-spectra and consider the corresponding graphs.
... In the following theorem some exact expressions for the first four spectral moments of the Q-spectrum of G are given. 4,12]). Let G be a graph with n vertices, m edges, N G (C 3 ) triangles and degree sequence deg ...
... (2) d 1 (H) ≥ p + 3. Then by (12) ...
If q copies of K 1,p and a cycle C q are joined by merging any vertex of C q to the vertex with maximum degree of K 1,p , then the resulting graph is called the jellyfish graph JF G(p, q) with parameters p and q. Two graphs are said to be Q-cospectral (respectively, L-cospectral) if they have the same signless Laplacian (respectively, Laplacian) spectrum. A graph is said to be DQS (respectively, DLS) if there is no other non-isomorphic graphs Q-cospectral (respectively, L-cospectral) with it. In [M. Mirzakhah and D. Kiani, The sun graph is determined by its signless Laplacian spectrum, Electron J. Linear Algebra, 20 (2010) 610-620] it were proved that the sun graphs are DQS, where Q(G) is used for the signless Laplacian matrix of G. Additionally, in [R. Boulet, Spectral characterizations of sun graphs and broken sun graphs, Discrete Math. Theor. Comput. Sci. 11 (2) (2009) 149-160] it was proved that the sun graphs are also DLS, where L(G) denotes the Laplacian matrix of G. In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS (for when q is an even number) and DQS.
... posed the question Which graphs have integral spectra?, the search for A-integral graphs or L-integral graphs has been done. More recently, Q-integral graphs were introduced in the literature [1,7,8,9,4]. ...
... In the year 1974, Harary and Schwenk [13] introduced the concept of integral graphs. Several results on these graphs can be found in [1,2,4,15,17,21]. It is observed that CCC(G) is super integral for the groups mentioned above. ...
The commuting conjugacy class graph of a non-abelian group $G$, denoted by $\mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $\mathcal{CCC}(G)$ is super integral for these groups. We compare various energies and as a consequence it is observed that $\mathcal{CCC}(G)$ satisfy E-LE Conjecture of Gutman et al. We also provide negative answer to a question posed by Dutta et al. comparing Laplacian and Signless Laplacian energy. Finally, we conclude this paper by characterizing the above mentioned groups $G$ such that $\mathcal{CCC}(G)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.
... Following our idea of [5], we establish an existence condition for bipartite r-regular graphs with q = 0. Namely, the adjacency matrix of such a graph can be written in the form ...
A signed graph is called integral if its spectrum consists entirely of integers, it is r-regular if its underlying graph is regular of degree r, and it is net-balanced if the difference between positive and negative vertex degree is a constant on the vertex set (this constant is called the net-balance and denoted d). We determine all the connected integral 3-regular net-balanced signed graphs. In the next natural step, for r = 4, we consider only those whose net-balance is a simple eigenvalue. There, we complete the list of feasible spectra in bipartite case for = 0 and prove the non-existence for d= 0. Certain existence conditions are established and the existence of some 4-regular (simple) graphs is confirmed. In this study we transferred some results from the theory of graph spectra; in particular, we give a counterpart to the Hoffman polynomial.
... By Lemma 3.5, there exists a subgraph G 1 of H j such that G 1 ∼ = JF G(p, q ′ ). By Corollary 3.2, q 1 (G) = q 1 (G 1 ) = p + 5 + p 2 + 6p + 13 2 . On the other hand, it follows from Lemma 2.5 that q 1 (H j ) > q 1 (G 1 ). ...
In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS and DQS.
... One may conf. [3,6,1,19,21,27] for various results of these graphs. ...
In this paper, we obtain energy, Laplacian energy and signless Laplacian energy of the commuting graphs of some families of finite non-abelian groups.
... Lemma 4.4[24,25] -Let G be graph with n vertices, m edges, N G (C 3 ) triangles anddeg(G) = (d 1 , d 2 , . . . , d n ). ...
Let G
s,t
denote the wind-wheel graph on n vertices obtained by appending s triangle(s) to a pendant vertex of a path P
t+1 with just a vertex in common. In this paper, we prove that all wind-wheel graphs are determined by their Laplacian spectra as well as signless Laplacian spectra. As G
s,t
is the well-known friendship graph if t = 0, our results include that the friendship graph is determined by its Laplacian spectrum as well as signless Laplacian spectrum, which provides of a new proofs of the results in [15].
... The study on integral graphs and Q-integral graphs has drawn many scholars' attentions. Results about them are found in [5, 7, 8, 13-16, 22, 26, 32] and [10,23,29,34], respectively. ...
Let D(G) = (dij)n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vi in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In this paper, we investigate distance integral complete r-partite graphs Kp1,p2,...,pr = Ka1.p1,a2.p2,...,as.ps and give a sufficient and necessary condition for Ka1.p1,a2.p2,...,as.ps to be distance integral, from which we construct infinitely many new classes of distance integral graphs with s = 1,2,3,4. Finally, we propose two basic open problems for further study
... [6,17]. Other work can be found in [16,34] for the spectral integrality, [29,35] for the isospectral problem, and [28] for the spectral spread. For more papers, one can refer to [14] and the references therein. ...
Let B(n,r) be the set of all bicyclic graphs with n vertices and r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n,r).
... posed the question Which graphs have integral spectra?, the search for A-integral graphs or L-integral graphs has been done. More recently, Q-integral graphs were introduced in the literature [1,7,8,9,4]. We see however that in many applications, just the index needs to be an integer. ...
Let G be a simple graph and A=A(G), L=L(G) and Q=Q(G) the adjacency, the Laplacian and the signless Laplacian matrices of G, respectively. For each of the associated matrices of G, M=A, L, or Q, we call M-spectrum of G the spectrum of the matrix M. In this work we present the KK n j graphs, obtained from two copies of the complete graph K n by adding j edges, 1≤j≤n, between a vertex of one of the copies and j vertices of the other. We obtain M-spectral properties of this graph based on its clique number and its edge connectivity.
... They are usually called the signless Laplacian eigenvalues of G . The matrix Q (G) is well studied by several authors [1][2][3][4][5][6][7][8][10][11][12][13]. In [5], Cvetković et al. gave the following conjecture involving the second largest signless Laplacian eigenvalue and the index of graph G (see also [1]). ...
Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In [5], Cvetković et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q2) and the index (λ1) of graph G (see also Aouchiche and Hansen [1]):1-n-1⩽q2-λ1⩽n-2-2n-4with equality if and only if G is the star K1,n-1 for the lower bound, and if and only if G is the complete bipartite graph Kn-2,2 for the upper bound. In this paper we prove the lower bound and characterize the extremal graphs.
... Let A(G) = (a ij ), where [7] posed the question Which graphs have integral spectra?, the search for A-integral graphs or L-integral graphs has been done. More recently, Q-integral graphs were introduced in the literature [2,6,11,12,13]. We see however that in many applications, just the M -index (the greatest M -eigenvalue) needs to be an integer. The indices with respect to the three matrices considered, A, L and Q, are parameters related to several other graph invariants, such as chromatic number, maximum degree, average degree and independence number [3]. ...
Let G be a simple graph and A = A(G), L = L(G) and Q = Q(G) the adjacency, the Laplacian and the signless Laplacian matrices of G, respectively. For each of the associated matrices of G, M = A, L or Q, we call M-spectrum of G the spectrum of the matrix M. In this work we present the KKj^n graphs, obtained from two copies of the complete graph Kn by adding j edges, 1 <= j <= n, between a vertex of one of the copies and j vertices of the other. We obtain M-spectral properties of this graph based on its clique number and its edge connectivity. Besides that we investigate conditions for the M-index (the largest M-eigenvalue) to be an integer number.
... The least signless Laplacian eigenvalues is also studied; see e.g., [6,11]. Other work can be found in [15,23] for the spectral integrality, [21,22] for the isospectral problem, and [20] for the spectral spread. The signless Laplacian matrix of a graph can be considered as a special case of the Laplacian matrix of a mixed graph with all edges unoriented; see [2,11,13,29] for the details. ...
In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity.
... Only recently has the signless Laplacian attracted the attention of researchers. As our bibliography shows, several papers on the signless Laplacian spectrum (in particular, [2], [5], [10], [12], [14], [15], [16], [24], [26], [28], [29], [30], [31], [32], [35], where the signless Laplacian is explicitly used) have been published since 2005. We are also aware that several other papers are being prepared, or are already in the process of publication. ...
A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix �� which is in a prescribed way defined for any graph. This theory is called �� -theory. We outline a spectral theory of graphs based on the signless Laplacians �� and compare it with other spectral theories, in particular with those based on the adjacency matrix �� and the Laplacian �� . The �� -theory can be composed using various connections to other theories: equivalency with �� -theory and �� -theory for regular graphs, or with �� -theory for bipartite graphs, general analogies with �� -theory and analogies with �� -theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.
... [6,27] Let G be a graph with n vertices, m edges and n 3 (G) triangles. Let T k = n i=1 ν k i be the kth Q-spectral moment of G, k = 0, 1, 2, . . ...
A propeller graph is obtained from an $\infty$-graph by attaching a path to
the vertex of degree four, where an $\infty$-graph consists of two cycles with
precisely one common vertex. In this paper, we prove that all propeller graphs
are determined by their Laplacian spectra as well as their signless Laplacian
spectra.
... The matrix Q (G) seems to be less well known. The matrix Q (G) is well studied by several authors [1][2][3][4][5][6]8,15,26,29,34,35,32]. ...
Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.
... Since the Laplacian spectrum of the path P n is 2 + 2 cos iπ n (i = 1, 2,. .. , n), Lemma 2.3 implies that the Q-spectrum of path P n is also 2 + 2 cos iπ n (i = 1, 2,. .. , m). By Lemma 2.7, ν 2 (H n,p ) ≤ ν 1 (P n ) < 4. Lemma 2.9 ( [3,16]). Let G be a graph with n vertices, m edges, t triangles and vertex degrees d 1 ...
A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q, any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. In this paper, it is proved that all lollipop graphs are determined by their Q-spectra.
... Let G be a graph with V (G) ̸ = ∅ and E(G) ̸ = ∅. Then Lemma 2.8 ([7,21]). Let G be a graph with n vertices, m edges, n G (C 3 ) triangles and deg(G) = (d 1 , d 2 , . . . ...
Let H(n;q,n1,n2) be a graph with n vertices containing a cycle Cq and two hanging paths Pn1 and Pn2 attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12;6,1,5) and H(12;8,2,2), no two non-isomorphic graphs of the form H(n;q,n1,n2) are A-cospectral. It is proved that all graphs H(n;q,n1,n2) are determined by their L-spectra. And all graphs H(n;q,n1,n2) are proved to be determined by their Q-spectra, except for graphs H(2a+4;a+3,a2,a2+1) with a being a positive even number and H(2b;b,b2,b2) with b≥4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.
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.
A connected Q-integral graphs with Q-spectral radius 6 is either known or has maximum edge-degree 5 or is a bipartite graph containing S3,3 as an induced subgraph, where S3,3 is the double star which is obtained from two disjoint claws K1,3 by adding one edge between the two vertices of degree 3. In this paper, we will slightly improve this result by showing that there is no connected Q-integral bipartite graph with Q-spectral radius 6 containing S3,3 as an induced subgraph.
A graph G is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum as G. Let k, t i (1 ≤ i ≤ k) and s be natural numbers. A path-friendship graph, G s,t1,··· ,t k , is a graph of order n = 2s + t 1 + · · · + t k + 1 which consists of s triangles and k paths of lengths t 1 , t 2 ,. .. , t k sharing a common vertex. In this paper, we show that these graphs are DQS and using this result we respond to a conjecture in (F. Wen, Q. Huang, X. Huang and F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure and Appl. Math., 46 (2015), 613-631).
Let G a graph on n vertices. The signless Laplacian matrix of G, denoted by Q(G), is defined as Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the degrees of G. A graph G is said to be Q-integral if all eigenvalues of the matrix Q(G) are integers. In this paper, we characterize all Q-integral graphs among all connected graphs with at most two vertices of degree greater than or equal to three.
Q-integral graphs are graphs whose signless Laplacians have only integral eigenvalues. The edge-degree of an edge is the number of edges adjacent to the edge. In 2008, S. K. Simić and Z. Stanić studied connected Q-integral graphs with edge-degrees at most five. In this paper, we study connected Q-integral graphs with edge-degrees at most six.
A graph is called Q-integral if its signless Laplacian spectrum consists of integers. In this paper, we characterize a class of k-cyclic graphs whose second smallest signless Laplacian eigenvalue is less than one. Using this result we determine all the Q-integral unicyclic, bicyclic and tricyclic graphs.
A finite non-abelian group $G$ is called super integral if the spectrum, Laplacian spectrum and signless Laplacian spectrum of its commuting graph contain only integers. In this paper, we first compute various spectra of several families of finite non-abelian groups and conclude that those groups are super integral. As an application of our results we obtain some positive integers $n$ such that $n$-centralizer groups are super integral. We also obtain some positive rational numbers $r$ such that $G$ is super integral if it has commutativity degree $r$. In the last section, we show that $G$ is super integral if $G$ is not isomorphic to $S_4$ and its commuting graph is planar. We conclude the paper showing that $G$ is super integral if its commuting graph is toroidal.
In this paper, we study the Drazin inverse of P+Q, where P is idempotent, and derive additive formulas under condition PQ(2)=0 or Q(2)P=0. As its applications we establish some representations for the Drazin inverse of a class of block matrices with an idempotent subblock and under some conditions expressed in terms of the individual blocks. The results extend earlier work obtained by several authors.
In this article, we shall give the adjacency characteristic polynomial, Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of the join-based compositions of arbitrary graphs in terms of the corresponding characteristic polynomial of G(i)(i = 1, 2, 3). These characterizations allow us to exhibit many infinite families of integral graphs.
Let G be a simple graph. The matrix Q(G) = D(G) + A(G) is the signless Laplacian matrix of G, where D(G) and A(G) is the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. The Laplacian matrix of G is the matrix L(G) = D(G)-A(G). A graph iscalled L-integral (resp. Q-integral) if its (signless) Laplacian spectrum consists entirely of integers. Let G 1 and G2 be two graphs, and S(G) be the subdivision graph of G. Then the Svertex join of G1 and G2, denoted by G1 V G2 is obtained from S(G1 ) and G2 by joining each vertices of G1 to each vertices of G2. The Sedge join of G1 with G2, denoted by G1 VG2 is obtained from S(G1) and G2 by joining all vertices of S(G1) corresponding to the edges of G1 with all vertices of G2. In this paper we obtain the Q-spectra and L-spectra of these two joins of graphs when G 1 and G2 are regular graphs. As an application, some infinite families of Q-integral graphs and L-integral graphs are obtained.
For a graph G of order n, the signless Laplacian matrix of G is Q(G)=D(G)+A(G), where A(G) is its adjacency matrix and D(G) is the diagonal matrix of the vertex degrees in G. The signless Laplacian characteristic polynomial (or Q-polynomial) of G is QG(x)=|x In-Q(G)|, where In is the n×n identity matrix. A graph G is called Q-integral if all the eigenvalues of its signless Laplacian characteristic polynomial QG(x) are integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be Q-integral, from which we construct infinitely many new classes of Q-integral graphs. Finally, we propose two basic open problems for further study.
The authors' monograph Spectral Generalizations of Line Graphs was published in 2004, following the successful use of star complements to complete the classification of graphs with least eigenvalue -2. Guided by citations of the book, we survey progress in this area over the past decade. Some new observations are included.
A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.
A graph is Q-integral if the spectrum of its signless Laplacian matrix consists entirely of integers. In their study of Q-integral complete multipartite graphs, G. Zhao et al. [Linear Algebra Appl. 438, No. 3, 1067–1077 (2013; Zbl 1257.05091)] posed two questions on the existence of such graphs. We resolve these questions and present some further results characterizing particular classes of Q-integral complete multipartite graphs.
For a simple undirected graph G, denote by A(G) the (0,1)-adjacency matrix of G. Let thematrix S(G) = J-I-2A(G) be its Seidel matrix, and let S
G
(λ) = det(λI-S(G)) be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of S
G
(λ) are integral, then the graph G is called S-integral. In this paper, our main goal is to investigate the eigenvalues of S
G
(λ) for the complete multipartite graphs G = \(G = K_{n_1 ,n_2 ,...n_t } \). A necessary and sufficient condition for the complete tripartite graphs K
m,n,t
and the complete multipartite graphs
to be S-integral is given, respectively.
Graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure. We show that regular, bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles. This enables to determine, from the spectrum alone, the feasible families of numbers of common neighbors for each vertex with other vertices in its part. For particular spectra, such as [6,29,06,-29,-6] (where exponents denote eigenvalue multiplicities), there is a unique such family, which makes it possible to characterize all graphs with this spectrum.Using this lemma we also to show that, for r⩾2, a graph has spectrum [r,rr(r-1),02(r-1),-rr(r-1),-r] if and only if it is a graph of a 1-resolvable transversal design TD(r,r), i.e., if it corresponds to the complete set of mutually orthogonal Latin squares of size r in a well-defined manner.
This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results.
A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.
Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In [Publ. Inst. Math., Nouv. Sér. 81(95), 11–27 (2007; Zbl 1164.05038)], C. Cvetković, P. Rowlinson and S. K. Simić have given conjectures on signless Laplacian eigenvalues of G (see also M. Aouchiche and P. Hansen [Linear Algebra Appl. 432, No. 9, 2293–2322 (2010; Zbl 1218.05087)], C. S. Oliveira et al. [Linear Algebra Appl. 432, No. 9, 2342–2351 (2010; Zbl 1214.05082)]). Here we prove two conjectures.
In this paper, we classify the connected non-bipartite integral graphs with
spectral radius three.
A graph is called integral if the spectrum of its adjacency matrix consists entirely of integers. A class of non-regular bipartite integral graphs with maximum degree four has been characterized by generating all its elements. An exact algorithm for generating these graphs is described; tables containing some invariants and figures showing their structures are given.
The problem of identifying those graphs whose spectra consist entirely of integers was first posed by F. Harary. We examined some elementary procedures for constructing integral graphs in [6]. Although the general problem seems intractible, it is easy to find the seven connected graphs with integral spectra, maximum degree at most three, and minimum degree less than three. This article was inspired by Cvetković's attempt [4] to find the connected cubic integral graphs. He had displayed twelve such graphs, and had restricted the remaining possibilities to ninety-five potential spectral.
In this article we construct the sole graph omitted from Cvetković's list and prove that no other exist. We have just learned that Cvetković has recently collaborated with Bussemaker [1] to obtain the same result. Unlike their effort, the present article avoids the use of computer search to examine all the possibilities.
The thirteenth graph happens to have the same spectrum as one of the others. This cospectral pair confirms a conjecture of Balaban by being indistinguishable under a certain proposed chemical classification scheme.
Line graphs have the property that their least eigenvalue is greater than or equal to –2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.
An integral graph is a graph whose spectrum is integral. By this paper we start finding all integral graphs with the maximum vertex degree four. Particularly, we find those of them which are nonregular and nonbipartite.
Using graph angles and previous results from [8] we show that a connected bipar-tite 4-regular integral graph has at most 1260 vertices, except if it has one of five exceptional spectra.
The graphs 1–7, and only these molecular graphs have integral spectra. The proof of this theorem elucidates also several other interesting spectral properties of graphs which represent unsaturated conjugated compounds.
Possible spectra of 4-regular integral graphs are determined. Some constructions and a list of 65 known connected 4-regular integral graphs are given.
We determine all connected 4-regular integral graphs avoiding ±3 in the spectrum. There are exactly 16 bipartite and 8 nonbipartite such graphs. The smallest bipartite one is K4,4, while the largest has 32 vertices. Among these graphs there are two triplets of cospectral nonisomorphic graphs and two pairs of cospectral nonisomorphic graphs. The smallest nonbipartite one is K5, and the largest has 15 vertices. Among these graphs there is a pair of cospectral nonisomorphic graphs.
We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements.
For most feasible spectra of connected regular graphs with four distinct eigenvalues and at most 30 vertices we find all such graphs, using both theoretic and computer results.
Simi´ Laplacian of finite graphs
- D Cvetkovi´
D. Cvetkovi´, S. Simi´ Laplacian of finite graphs, Linear Algebra Appl. 423 (2007) 155–171.
There are 93 non-regular, bipartite integral graphs with maximum degree four
- M Lepović
- S K Simić
- K T Balińska
- K T Zwierzyński
M. Lepović, S.K. Simić, K.T. Balińska, K.T. Zwierzyński, There are 93 non-regular, bipartite integral graphs with maximum degree four, CSC
Report No. 511, Technical University of Poznań, 2005, pp. 1-17.