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On the spectral radius of unicyclic graphs with prescribed degree sequence
Abstract
We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence.
... Since then, similar problems have been studied extensively. The majorization theorems for (signless Laplacian) spectral radius of unicyclic graphs and bicyclic graphs were also discovered, and the unique ρ-maximal (μ-maximal) graph of (π) has been characterized for unicyclic and bicyclic degree sequences [9][10][11][12][13][14] (see Corollaries 6.3 and 6.4 of Section 6). ...
... Finally, if k = 2, u 1 = u 3 and there exists at least one pendant vertex not on the root tree with root u 2 , then by Lemma 3.11, we have u 1 u 3 ∈ E(H), and this completes the proof. Remark 3.14: In view of Proposition 1.1, Theorems 3.12 and 3.13 also hold for the A αmaximal graph in the case 0 ≤ α < 1. ρ β -maximal graph of (π, t; c) for c ∈ {0, 1, 2} The unique ρ-maximal graph (also the same unique μ-maximal graph) in the class of trees, unicyclic graphs and/or bicyclic graphs with given degree sequence has been determined in [7][8][9][10]13,14] (see Corollary 6.3), respectively. The uniqueness of the corresponding extremal tree has been proved [7,8]. ...
... The uniqueness of the corresponding extremal tree has been proved [7,8]. But for the cases of unicyclic graphs and bicyclic graphs, uniqueness of the corresponding extremal graph has not been given [9,10,13,14]. Furthermore, it is still an open problem the uniqueness of the ρ-maximal graph in the class of c-cyclic graphs with given degree sequence. ...
In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the universal adjacency matrix U=αA+βD+γJ+δI, where A, D, J, and I are the adjacency matrix of G, the degree matrix of G, the all-ones matrix, the identity matrix, respectively. Here, we consider Mβ=A+βD, with β≥0, and we study the graphs belonging to some given class Γ maximizing the corresponding spectral radius ρβ. In particular, we consider connected graphs with prescribed c-cyclic degree sequence, c∈{0,1,2}, and the multicone graphs defined over them, where the multicone graph is the join of a clique with a given graph. The aim of this paper is to provide the best possible generalization of results to the spectral radius of Mβ (and the graph matrices related to it) of several well-known results for multicone graphs over connected graphs with prescribed c-cyclic degree sequence, where c∈{0,1,2}.
... Lemma 1 (see [13]). Suppose that T is a simple connected graph with n vertices and maximum Δ, x � (x 1 , x 2 , . . . ...
... Lemma 2 (see [13]). Suppose that T is a simple connected graph with n vertices and maximum Δ, x � (x 1 , x 2 , . . . ...
... Lemma 3 (see [13]). Suppose that T � (V, E) is a simple connected graph with n vertices and maximum Δ, Figure 5; if T 1 is still a simple connected graph, then ρ(T 1 ) > ρ(T). ...
In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with n vertices and maximum degree Δ and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices n and maximum degree Δ from now on.
... Since then, similar problems have been studied extensively. The majorization theorems for (signless Laplacian) spectral radius of unicyclic graphs and bicyclic graphs were also discovered, and the unique ρmaximal (µ-maximal) graph of Γ(π) were characterized for unicyclic graph and bicyclic graph degree sequences π, respectively [1,6,7,11,15,19]. Here we have to point out that the majorization theorem of (signless Laplacian) spectral radius can not hold for all c-cyclic graphs, as counterexamples show that the majorization theorem for the (resp., signless Laplacian) spectral radius of 3-cyclic (resp., 4-cyclic) graphs does not hold [7,13]. ...
... Recently, Luo et al. [16] proved the majorization theorems for the (signless Laplacian) spectral radius of single-cone trees and single-cone unicyclic graphs respectively, but they characterize neither the ρ-maximal graphs nor the µ-maximal graphs of T (π, 1) and U (π, 1). Motivated by their research but further than that, in this paper, we first characterize the Θ α -maximal graphs of Γ(π, t; c), and then consider the majorization theorems for the general spectral radius of t-cone trees, t-cone unicyclic graphs and t-cone bicyclic graphs, respectively. ...
... , v 1,s 1 of the first layer such that they are adjacent to v 0,1 , and so d T π * (v 0,1 ) = s 1 = d * t+1 ; (iii) The remaining vertices of T π * appear in a BFS-connecting, that is: assume that all vertices of the p-th layer of T π * have been constructed and they are v p,1 , v p,2 , . . . , v p,sp ; by using the induction hypothesis, now we construct all the vertices of the (p + 1)-st layer; 1 , v p+1,2 , . . . , v p+1,s p+1 for the (p + 1)-st layer such that d T π * (v p,i ) − 1 vertices are adjacent to v p,i for i = 1, 2, . . . ...
The general spectral radius of a graph $G$, denoted by $\Theta(G,\alpha)$, is the maximal eigenvalue of $M_{\alpha}(G)=A(G)+\alpha D(G)$ $(\alpha\geq 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. A graph $G$ is called $\Theta_\alpha$-maximal in a class of connected simple graphs $\mathcal {G}$ if $\Theta(G,\alpha)$ is maximal among all graphs of $\mathcal {G}$. A $t$-cone $c$-cyclic graph is the join of a complete graph $K_t$ and a $c$-cyclic connected simple graph. Let $\pi=\big(d_1,d_2,\ldots,d_n\big)$ and $\pi'=\big(d'_1,d'_2,\ldots,d'_n\big)$ be two non-increasing degree sequences of $t$-cone $c$-cyclic graphs with $n$ vertices. We say $\pi$ is strictly majorized by $\pi'$, denoted by $\pi \lhd \pi'$, if $\pi\neq \pi'$, $\sum_{i=1}^n d_i=\sum_{i=1}^n d_i'$, and $\sum_{i=1}^k d_i\leq \sum_{i=1}^k d_i'$ for $k=1,2,\ldots,n-1$. Denote by $\Gamma(\pi,t;c)$ the class of $t$-cone $c$-cyclic graphs with $\pi$ as its degree sequence. In this paper, we determine some properties of $\Theta_\alpha$-maximal graphs of $\Gamma(\pi,t;c)$ and characterize the unique $\Theta_\alpha$-maximal graph of $\Gamma(\pi,t;0)$ \big(resp. $\Gamma(\pi,t;1)$ and $\Gamma(\pi,t;2)$\big). Moreover, we prove that if $\pi \lhd \pi'$, $G$ and $G'$ are the $\Theta_\alpha$-maximal graphs of $\Gamma(\pi,t;c)$ and $\Gamma(\pi',t;c)$ respectively, then $\Theta(G,\alpha)<\Theta(G',\alpha)$ for $c\in \big\{0,1\big\}$, and we also consider the similar result for $c=2$.
... In particular, R 0 −1 (G) in [28] is called the inverse degree ID(G) of G, R 0 2 (G) is just equal to M 1 (G), and R − 1 2 (G) in [22] is called the Randić 2 , as a generalization of M 1 (G), in [36] the authors defined the general sum-connectivity index χ a (G) of G as ...
... If π max is the degree sequence of a graph G ∈ G such that π ◁ π max holds for any other degree sequence π of these graphs in G, then π max is called the largest degree sequence of G. Proof. In view of (1.1), it is easy to see that π = (n − 1, 1 (n−1) ), π ′ = (n − 1, 2 (2) , 1 (n−3) ) and π ′′ = (n − 1, 3, 2 (2) , 1 (n−4) ) are the largest sequences among all the degree sequences of trees, unicyclic graphs and bicyclic graphs on n vertices, respectively. Furthermore, it is easy to check that ...
... Proof. Let π 1 = (α, 2 (n−α−1) , 1 (α) ), π 2 = (α + 1, 2 (n−α) , 1 (α−1) ), π 3 = (α, 3 (2) , 2 (n−α−3) , 1 (α) ), π 4 = (α + 1, 3 (2) , 2 (n−α−2) , 1 (α−1) ), π 5 = (α + 2, 2 (n−α+1) , 1 (α−2) ), and π 6 = (α, 4, 3 (2) , 2 (n−α−4) , 1 (α) ). Let π (c) max be the largest degree sequence among all the degree sequences of graphs in the class of c-cyclic graphs with n vertices and independence number α. ...
Identifying graphs with extremal properties is an extensively studied topic in both topological graph theory and spectral graph theory. As observed in the literature, for many graph categories the extremal graphs with respect to some prescribed topological index or spectral invariant are the same. Therefore, it is an interesting problem to find a unified method to identify the extremal graphs for a set of topological or spectral invariants. Here we consider the majorization theorem to obtain extremal graphs in the class of trees, unicyclic graphs and bicyclic graphs with given order and fixed maximum degree, independence number, matching number, domination number, and/or number of pendant vertices, respectively.
... In particular, R 0 −1 (G) in [28] is called the inverse degree ID(G) of G, R 0 2 (G) is just equal to M 1 (G), and R − 1 2 (G) in [22] is called 20 the Randić index R(G) of G. ...
... Let B be a bicyclic graph with degree sequence π and independence number α. Since α ≥ n 2 , and in view of 22 Theorem 3.3, the graphs Z 2 (n, α), Z 3 (n, α) and Z 4 (n, α) are extremal graphs of Γ (π 4 ), Γ (π 5 ) and Γ (π 6 ), respectively. ...
... Suppose that the degree sequence of U is π = (d 1 , d 2 , . . . , d n−p , 1 (p) ), where d 1 ≥ d 2 ≥ · · · ≥ d n−p ≥ 2 and C g is 22 the unique cycle of U, where g ≥ 3. By (5.1), we have p ≤ n − γ . ...
For a symmetric bivariable function f(x, y), let the connectivity function of a connected graph G be M f (G) = Puv ∈E(G ) f(d(u), d(v)), where d(u) is the degree of vertex u. As an application of majorization theory, we present a uniform method to some extremal results together with its corresponding extremal graphs for vertex-degree-based invariants among the class of trees, unicyclic graphs and bicyclic graphs with fixed number of independence number and/or matching number, respectively. As a consequence, several known results in chemical graph theory has been obtained. © 2019 University of Kragujevac, Faculty of Science. All rights reserved.
... Let U * π be the unique unicyclic graph in S(π), where v 1 , v 2 and v 3 are mutually adjacent to form C, the unique cycle of U * π , and the remaining vertices appear in BF S-ordering with respect to C starting from v 4 which is adjacent to v 1 [3,55]. ...
... Theorem 2.18. [3,55] Let π be a unicyclic degree sequence. Then, U * π is the unique maximal extremal unicyclic graph of S(π). ...
... Theorem 2.22 shows that the maximal extremal graphs of S(π) are BF S-graphs, but the BF S-graphs of S(π) are not unique. Thus, from Theorems 2.17-2.20, it is natural for us to consider the following question: Whether the maximal extremal graph of S(π) is unique for any c-cyclic degree sequence π? Actually, Belardo et al. [3] conjectured that the answer is positive. ...
Suppose π = (d1, d2, …, dn) and π′ = (d′ 1, d′ 2, …, d′ n) are two positive non- increasing degree sequences, write π ⊳ π′ if and only if π ≠ π′, (Equation presented)Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G′ be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with π and π′ as their degree sequences, respectively. If π ⊳ π′ can deduce that ρ(G) < ρ(G′) (respectively, μ(G) < μ(G′)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and G′ satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory. © 2015, International Linear Algebra Society. All rights reserved.
... In the last 5 years, the attention of many researchers has been attracted by the graphs with prescribed degree sequence, for which the maximizers are identified for trees [5], unicyclic graphs [4], and bicyclic graphs [11]. Tricyclic graphs are considered in the nice paper [10]. ...
... It is routine to check that ρ(θ 1 (8)) < 3.74 < b (8). Similarly, for 4 ≤ ∆ ≤ 7, we have ρ(θ 1 (7)) < 3.55 < b (7), ρ(θ 1 (6)) < 3.34 < b (6), ρ(θ 1 (5)) < 3.10 < b (5) and ρ(θ 1 (4)) < 2.828 < b (4). In view of Corollaries 3.3 and 4.3, the following chain of inequalities holds for any n ≥ 14 and 4 ≤ ∆ ≤ 8, (∆, n)). ...
... We use the same strategy of the previous lemma. Let θ 2 (∆) = θ m (∆, 11) = θ(∆, 4,4,4). We next show that ρ(θ 2 (∆)) < b(∆). ...
A simple connected non-regular graph is said to be -bidegreed, or biregular, if the vertices have degree from the set , with . We consider two classes of -bidegreed graphs denoted by and . A graph belongs to if: a) it is obtained from disjoint paths , , by identifying the vertices and the vertices , and the graph so obtained has n vertices; b) for each of the n vertices, pendant vertices are added, so that any vertex from any has degree . The class , , is similarly obtained by identifying all the vertices and from the ’s, into a single vertex. In this paper, we show that for any graph in or , the spectral radius of the adjacency matrix increases whenever the difference between the lengths of any two ’s increases. We also compute some bounds for the spectral radius when the lengths of the ’s tend to infinity. Finally, we discuss about bicyclic -bidegreed graphs with n degree vertices minimizing the spectral radius. We prove that in most cases such graphs do not belong to .
... In [3], the trees maximizing the spectral radius with prescribed degree sequences were determined by Bıyıkoglu and Leydold. Recently, Belardo et al. [4] investigated the case of unicyclic graphs. The results on some other classes of graphs can be seen, for example, in [12][13][14]18]. ...
... It is worth noticing that in the last section of [4] the authors gave a conjecture on the structure of the graph with maximal spectral radius in the set of connected graphs with prescribed degree sequence. In this paper, restricted to bicyclic graphs, we determine the maximal graphs with prescribed bicyclic degree sequences, which were predicted in the conjecture. ...
... Note that C is called the Perron-core of G in [4]. ...
Restricted to the bicyclic graphs with prescribed degree sequences, we determine the (unique) graph with the largest spectral radius with respect to the adjacency matrix.
... For Θ(d 1 , d 2 , . . . , d n ), the BSP has not been solved in general but has some results for some special graphs such as trees and unicyclic graphs [8,10]. Moreover, for the special cases of caterpillars [11] and cycles with spikes [12] the similar results to those of [8,10] also have been obtained. ...
... , d n ), the BSP has not been solved in general but has some results for some special graphs such as trees and unicyclic graphs [8,10]. Moreover, for the special cases of caterpillars [11] and cycles with spikes [12] the similar results to those of [8,10] also have been obtained. ...
... In particular, when G is connected, A(G) is irreducible and by the well-known Perron-Frobenius Theorem (see, for example, [21]), ρ(G) is simple and there is a unique positive unit eigenvector. We shall refer to such an eigenvector x as the adjacent Perron vector of G and denote the component corresponding to the vertex v of G by x v (x v is called ρ-weight of v with respect to x in [10]). ...
We determine the (unique) weighted tree with the largest spectral radius with respect to the adjacency and Laplacian matrix in the set of all weighted trees with a given degree sequence and positive weight set. Moreover, we also derive the weighted trees with the largest spectral radius with respect to the matrices mentioned above in the sets of all weighted trees with a given maximum degree or pendant vertex number and so on.
... Structural properties of unicyclic graphs have been characterized in several papers [20][21][22][23][24][25][26][27][28][29][30][31]. As an example, consider the n-vertex sun graphs denoted by SGn where n≥6 even integer. ...
... For the spectral radius of unicyclic graphs various upper bounds have been deduced [20][21][22][23][24][25][26][27][28][29][30][31]. ...
Starting with the study of the Collatz-Sinogowitz and the Albertson graph irregularity indices the relationships between the irregularity of graphs and their spectral radius are investigated. We also use the graph irregularity index defined as Ir(G) = Δ – δ, where Δ and δ denote the maximum and minimum degrees of G. Our observations lead to the answer for a question posed by Hong in 1993. The problem concerning graphs with the smallest spectral radius can be formulated as follows: If G is a connected irregular graph with n vertices and m edges, and G has the smallest spectral radius, is it true that Ir(G) =1? It will be shown that the answer is negative; counterexamples are represented by several cyclic graphs. Based on the previous considerations the problem proposed by Hong can be reinterpreted (refined) in the form of the following conjecture: If G is a connected irregular graph with n vertices and m edges, and G has the smallest spectral radius, then Ir(G)=1 if such a graph exists, and if not, then Ir(G)=2. Considering the family of unicyclic graphs for which Ir(G) ≥ 2, we prove that among n-vertex irregular unicyclic graphs the minimal spectral radius belongs to the uniquely defined short lollipop graphs where a pendent vertex is attached to cycle C n-1 . Moreover, it is verified that among n-vertex graphs there exists exactly one irregular graph J n having a maximal spectral radius and an irregularity index of Ir(J n )=1. Finally, it is also shown that by using the irregularity index Ir(G) a classification of n-vertex trees into (n-2) disjoint subsets can be performed. © 2018, Budapest Tech Polytechnical Institution. All rights reserved.
... The BSP for G π n if restricted on trees has been solved in [2]. Recently, Belardo et al. [3] solved the BSP for G π n if restricted on unicyclic graphs, and make the following general conjecture. ...
... Then ρ(G π max ) < ρ(G π ′ max ). Belardo et al. [3] characterize the unicyclic graph with maximum spectral radius among all unicyclic graphs with prescribed degree sequence. Using above results, following, we will provide a somewhat simpler method to determine the unicyclic graph which has maximum spectral radius among all unicyclic graphs with prescribed degree sequence. ...
In this paper, we first present the properties of the graph which maximize the spectral radius among all graphs with prescribed degree sequence. Using these results, we provide a somewhat simpler method to determine the unicyclic graph with maximum spectral radius among all unicyclic graphs with a given degree sequence. Moreover, we determine the bicyclic graph which has maximum spectral radius among all bicyclic graphs with a given degree sequence.
... Unicyclic graphs are the connected graphs with the same order and size. The spectral radius of unicyclic graphs were discussed in [4,[11][12][13][14][16][17][18]. In particular, Li and Feng [16] consider the minimal spectral radius of unicyclic graphs, and proposed the following conjecture: ...
In this paper, we consider the unimodality of the principal eigenvector of graphs. A unimodal lemma of the principal eigenvector on internal paths is obtained. This unimodal lemma is used to establish a cycle version of Li–Feng transformation with respect to the spectral radius. Another application of unimodal lemma is to determine the unicyclic graph with minimal spectral radius.
... This problem is well studied in the literature for various classes of graphs, such as graphs with given diameter [10], edge chromatic number [4], domination number [23], unicyclic graphs with prescribed degree sequence [1]. One can refer to the recent book by Stevanović [22]. ...
... Biyikoglu et al. [2] determined the unique tree with maximum A 0 -spectral radius in the set of all trees with prescribed degree sequence. Belardo et al. [3] determined the (unique) graphs with the largest A 0 -spectral radius in the set of all unicyclic graphs with prescribed degree sequence. ...
Let $G$ be a graph with adjacency matrix $A(G)$, and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in[0,1]$, write $A_\alpha(G)$ for the matrix $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G).$$ This paper presents some extremal results about the spectral radius $\rho(A_\alpha(G))$ of $A_\alpha(G)$ that generalize previous results about $\rho(A_0(G))$ and $\rho(A_{\frac{1}{2}}(G))$. In this paper, we give some results on graph perturbation for $A_\alpha$-matrix with $\alpha\in [0,1)$. As applications, we characterize all extremal trees with the maximum $A_\alpha$-spectral radius in the set of all trees with prescribed degree sequence firstly. Furthermore, we characterize the unicyclic graphs that have the largest $A_\alpha$-spectral radius for a given unicycilc degree sequence.
... This problem is well studied in the literature for various classes of graphs, such as graphs with given diameter [10], edge chromatic number [4], domination number [23], unicyclic graphs with prescribed degree sequence [1]. One can refer to the recent book by Stevanović [22]. ...
The deficiency of a graph G, denoted by def(G), is the number of vertices unmatched under a maximum matching in G. We call G β-deficient if def(G) ≤ β. In this paper, by making use of the Q-spectral radius of a graph G, we present a sufficient condition for G with fixed minimum degree to be β-deficient, when G has large order.
... An upper bound of the spectral radius in double-star-like tree systems with maximal degree 4 was given in YZ Fan et al. 14 Since unicyclic graphs contain exactly one cycle, which closely relates with trees, many researchers study the spectral radius of the unicyclic graphs. The specific spectral radius of the unicyclic graphs with given degree sequences was studied in F Belardo et al. 15 Bounds of spectral radius of unicylic graphs are discussed in previous studies. [16][17][18] The upper bound of the Laplacian spectral radius was presented in Zhou and Xu 19 using the Cauchy-Schwarz inequality, Li and Feng 20 gave the maximum eigenvalue of graphs, and Hou and Li 21 mainly studied the bounds of the spectral radius of the tree graphs with some certain matching. ...
In this article, we research on the spectral radius of extremal graphs for the unicyclic graphs with girth g mainly by the graft transformation and matching and obtain the upper bounds of the spectral radius of unicyclic graphs.
... The signless Laplacian spectral radius of G is the largest eigenvalue of Q ( G ), denoted by μ( G ). The spectral radius of a graph has been studied extensively, see, e.g., [1,2,4,5,8,17,21,23,24] . In recent years, the signless Laplacian spectral radius has also received much attention, see, e.g., [13,16,19,26,27] . ...
The unique graphs with maximum spectral radius and signless Laplacian spectral radius are determined among trees, unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices (vertices of degree at least 3) using a unified approach.
... for any tree T ∈ T (2) n,∆ , with equality if and only if T = T * π . ...
This paper surveys some recent results and progress on the extremal prob-
lems in a given set consisting of all simple connected graphs with the same
graphic degree sequence. In particular, we study and characterize the extremal
graphs having the maximum (or minimum) values of graph invariants such as
(Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first
Dirichlet eigenvalue, the Wiener index, the Harary index, the number of
subtrees and the chromatic number etc, in given sets with the same tree,
unicyclic, graphic degree sequences. Moreover, some conjectures are included.
... In the last 5 years, the attention of many researchers has been attracted by the graphs with prescribed degree sequence, for which the maximizers are identified for trees [5], unicyclic graphs [4], and bicyclic graphs [11]. Tricyclic graphs are considered in the nice paper [10]. ...
A graph is said to be (Delta,delta)-bidegreed if vertices all have one of two possible degrees: the maximum degree Delta or the minimum degree delta, with Delta not equal delta. We show that in the set of connected (Delta,1)-bidegreed graphs, other than trees, with prescribed degree sequence, the graphs minimizing the adjacency matrix spectral radius cannot have vertices adjacent to Delta - 1 vertices of degree 1, that is, there are not any hanging trees. Further we consider the limit point for the spectral radius of (Delta,1)-bidegreed graphs when degree Delta vertices are inserted in each edge between any two degree Delta vertices.
... In this line, the unique extremal graph of Γ(π) was characterized when Γ(π) are restricted on trees, unicyclic graphs and/or bicyclic graphs, respectively [1] [2] [5] [11] [16] [17], and the (signless Laplacian) spectral radii of extremal graphs were proved to satisfy the majorization theorem when Γ(π) are restricted on trees, unicyclic graphs and/or bicyclic graphs, respectively [2] [5] [6] [8] [16] [17]. Furthermore, Liu et al. [9] found that the majorization theorem is a good tool to deal with Cvetkovi´c's problem, asked how to classify and order graphs according to their spectral radii [3]. ...
Some new properties are presented for the extremal graphs with largest (signless Laplacian) spectral radii in the set of all connected graphs with prescribed degree sequences, via which we determine all the extremal tricyclic graphs in the class of connected tricyclic graphs with prescribed degree sequences, and we also prove some majorization theorems of tricyclic graphs with special restrictions.
... until all vertices of G are processed. Such an ordering is called a spiral like ordering [2], or spiral like disposition [1,6]. Let F 1 (n, ∆), F 2 (n, ∆), and J 1 (n, ∆) be the unicyclic graphs with n vertices and the maximum degree ∆ ≥ n 2 + 1 as shown in Fig. 3.1. ...
Let Δ(G), Δ for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree Δ, where Δ≥n+1 2 (resp., Δ≥n 2+1 and Δ≥n+3 2) are determined. Moreover, it is shown that the spectral radius of a unicyclic graph U (resp., a bicyclic graph B) on n vertices strictly increases with its maximum degree when Δ(U)≥1 91+6n+10 2 (resp., Δ(U)≥1 92+6n+28 2 ).
... A graph is unicyclic if it has a unique cycle. Unicyclic graphs keep enjoying plenty of interest, as one can see, for instance, in [3,23,26,27,31,32]. ...
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.
... Let U * π be the unique unicyclic graph in Γ (π ), where v 1 , v 2 and v 3 are mutually adjacent, and form C , the unique cycle of U * π , and the remaining vertices appear in spiral like disposition with respect to C starting from v 4 that is adjacent to v 1 [12]. Theorem 2.1 ([12]). ...
This paper presents a simple method to order trees, unicyclic graphs and bicyclic graphs according to their spectral radii, which is based on the application of the “majorization theorem” of tree, unicyclic and bicyclic graphs, respectively. Moreover, we obtain some new results on the order of spectral radii of unicyclic and bicyclic graphs by employing this new method.
... Unicyclic graphs keep enjoying plenty of interest, as one can see , for instance, in [1], [4], [6], [11], [17], [19], [20]. ...
A set S is independent in a graph G if no two vertices from S are adjacent.
The independence number alpha(G) is the cardinality of a maximum independent
set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|,
then G=(V,E) is called a Konig-Egervary graph. The number
d_{c}(G)=max{|A|-|N(A)|} is called the critical difference of G (Zhang, 1990).
By core(G) (corona(G)) we denote the intersection (union, respectively) of all
maximum independent sets, while by ker(G) we mean the intersection of all
critical independent sets. A connected graph having only one cycle is called
unicyclic. It is known that ker(G) is a subset of core(G) for every graph G,
while the equality is true for bipartite graphs (Levit and Mandrescu, 2011).
For Konig-Egervary unicyclic graphs, the difference |core(G)|-|ker(G)| may
equal any non-negative integer. In this paper we prove that if G is a
non-Konig-Egervary unicyclic graph, then: (i) ker(G)= core(G) and (ii)
|corona(G)|+|core(G)|=2*alpha(G)+1. Pay attention that
|corona(G)|+|core(G)|=2*alpha(G) holds for every Konig-Egervary graph.
... Unicyclic graphs keep enjoying plenty of interest, as one can see, for instance, in [1], [3], [7], [15], [18], [20], [21]. ...
A set S is independent in a graph G if no two vertices from S are adjacent.
By core(G) we mean the intersection of all maximum independent sets. The
independence number alpha(G) is the cardinality of a maximum independent set,
while mu(G) is the size of a maximum matching in G. A connected graph having
only one cycle, say C, is a unicyclic graph. In this paper we prove that if G
is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G)
coincides with the union of cores of all trees in G-C.
The deficiency of a graph G, denoted by def(G), is the number of vertices unmatched under a maximum matching in G. We call G β-deficient if def(G)≤β. In this paper, by making use of the Q-spectral radius of a graph G, we present a sufficient condition for G with fixed minimum degree to be β-deficient, when G has large order.
For any real α ∈ [0, 1], Aα(G)=αD(G)+(1−α)A(G) is the Aα-matrix of a graph G, where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the degrees of G. This paper presents some extremal results about the spectral radius λ1(Aα(G)) of Aα(G) that generalize previous results about λ1(A0(G)) and λ1(A12(G)). In this paper, we study the behavior of the Aα-spectral radius under some graph transformations for α ∈ [0, 1). As applications, we show that the greedy tree has the maximum Aα-spectral radius in GD when D is a tree degree sequence firstly. Furthermore, we determine that the greedy unicyclic graph has the largest Aα-spectral radius in GD when D is a unicyclic graphic sequence, where GD={G∣GisconnectedwithDasitsdegreesequence}.
Octagonal systems are tree-like graphs comprised of octagons that represent a class of polycyclic conjugated hydrocarbons. In this paper, a roll-attaching operation for the calculation of the characteristic polynomials of octagonal chain graphs is proposed. Based on these characteristic polynomials, the extremal octagonal chains with n octagons having the maximum and minimum spectral radii are identified.
In this paper we consider bidegreed graphs, namely graphs whose vertices have degree either Δ or δ, with Δ≠δ. In the latter setting we identify those unicyclic graphs minimizing the largest eigenvalue of the adjacency matrix. We also give some remarks on the general case and a conjecture on the bicyclic case.
Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees. Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the readers research. • Dedicated coverage to one of the most prominent graph eigenvalues • Proofs and open problems included for further study • Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem.
Let G = (V (G), E(G)) be a simple connected graph of order n. For any vertices u, v,w ∈ V (G) with uv ∈ E(G) and uw ∈ E(G), an edge-rotating of G means rotating the edge uv (around u) to the non-edge position uw. In this work, we consider how the least eigenvalue of a graph perturbs when the graph is performed by rotating an edge from the shorter hanging path to the longer one. © 2015, Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.
Let (Formula presented.) denote the maximum size of an independent set of vertices and (Formula presented.) be the cardinality of a maximum matching in a graph (Formula presented.). A matching saturating all the vertices is a perfect matching. If (Formula presented.), then (Formula presented.) is called a König–Egerváry graph. A graph is unicyclic if it is connected and has a unique cycle. It is known that a maximum matching can be found in (Formula presented.) time for a graph with (Formula presented.) vertices and (Formula presented.) edges. Bartha (Proceedings of the 8th joint conference on mathematics and computer science, Komárno, Slovakia, 2010) conjectured that a unique perfect matching, if it exists, can be found in (Formula presented.) time. In this paper we validate this conjecture for König–Egerváry graphs and unicylic graphs. We propose a variation of Karp–Sipser leaf-removal algorithm (Karp and Sipser in Proceedings of the 22nd annual IEEE symposium on foundations of computer science, 364–375, 1981) , which ends with an empty graph if and only if the original graph is a König–Egerváry graph with a unique perfect matching (obtained as an output as well). We also show that a unicyclic non-bipartite graph (Formula presented.) may have at most one perfect matching, and this is the case where (Formula presented.) is a König–Egerváry graph.
This work investigates how the least eigenvalue of a graph behaves when the graph is perturbed by grafting an edge.
The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. In this article, we identify the (unique) weighted unicyclic graph with the largest spectral radius among the set of all n-vertex weighted unicyclic graphs with a fixed weight set W n = {w 1, w 2, … , w n }, where w 1 = w 2 = ··· = w n > 0 or, w 1 w 2 ··· w n > 0 and .
The Faber-Krahn inequality states that the ball has minimal first Dirichlet
eigenvalue among all bounded domains with the fixed volume in $\mathbb{R}^n$.
In this paper, we investigate the similar inequality for unicyclic graphs. The
results show that the Faber-Krahn type inequality also holds for unicyclic
graphs with a given graphic unicyclic degree sequence with minor conditions.
Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgraphs.
We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization.
We investigate the relationship between the structure of a graph and its eigenspaces. The angles between the eigenspaces and the vectors of the standard basis of n play an important role. The key notion is that of a special basis for an eigenspace called a star basis. Star bases enable us to define a canonical bases of n associated with a graph, and to formulate an algorithm for graph isomorphism.
We consider the following two classes of simple graphs: open necklaces and closed necklaces, consisting of a finite number of cliques of fixed orders arranged in path-like pattern and cycle-like pattern, respectively. In these two classes we determine those graphs whose index (the largest eigenvalue of the adjacency matrix) is maximal.
KeywordsAdjacency spectrum-Signless Laplacian spectrum-Caterpillars-Unicyclic graphs-Line graphs-Largest eigenvalue
Mathematics Subject Classification (2000)05C50
We identify in some classes of unicyclic graphs (of fixed order and girth) those graphs whose index, i.e. the largest eigenvalue, is maximal. Besides, some (lower and upper) bounds on the indices of the graphs being considered are provided.
The index of a graph is the largest eigenvalue of its adjacency matrix. Among the trees with a fixed order and diameter, a graph with the maximal index is a caterpillar. In the set of caterpillars with a fixed order and diameter, or with a fixed degree sequence, we identify those whose index is maximal.
Spectra of Graphs – Theory and Applications, III revised and enlarged ed
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[5] D. Cvetkoví
c, P. Rowlinson, S. Simí
c, Eigenspaces of Graphs, Cambridge University Press, 1997.
Graphs with given degree sequence and maximal spectral radius
- Biyukoglu
T. Biyukoglu, J. Leydold, Graphs with given degree sequence and maximal spectral radius, Electron. J. Combin., 15 (2008)
#R119.