Laplacian Spectral Radius Of Graphs Research Articles

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1,$ respectively, which intersect in exactly one common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

(Distance) signless Laplacian spectral radius of graphs with fixed fractional matching number

(Distance) signless Laplacian spectral radius of graphs with fixed fractional matching number

On the [formula omitted]-spectral radius of graphs without linear forests

Let A(G) and D(G) be the adjacency and degree matrices of a simple graph G on n vertices, respectively. The Aα-spectral radius of G is the largest eigenvalue of Aα(G)=αD(G)+(1−α)A(G) for a real number α∈[0,1]. In this paper, for α∈(0,1), we obtain a sharp upper bound for the Aα-spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough and characterize all graphs which attain the upper bound. As a result, we completely dertermine the maximum signless Laplacian spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough.

The maximum signless Laplacian spectral radius of graphs with forbidden subgraphs

The Brualdi-Solheid-Tur?n type problem has attracted wide attention recently. Let F be a set of graphs. A graph G is F -free if it does not contain any element of F as a subgraph. In this paper, when F1 = {W5,C6} and F2 = {F5}, we determine the maximum signless Laplacian spectral radius of Fi-free graph with n vertices, respectively and depict the corresponding extremal graph for i = 1,2.

Chromatic number and signless Laplacian spectral radius of graphs

Chromatic number and signless Laplacian spectral radius of graphs

On the Signless Laplacian Spectral Radius of Graphs without Small Books and Intersecting Quadrangles

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize all extremal graphs. In addition, we give an upper bound of the signless Laplacian spectral radius of all graphs which do not contain intersecting quadrangles as a subgraph.

Signless Laplacian spectral radius of graphs without short cycles or long cycles

The signless Laplacian spectral radius of a graph G, denoted by q(G), is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let G(m,g) be the family of graphs on m edges with girth g and H(m,c) be the family of graphs on m edges with circumference c. More precisely, we obtain the unique extremal graph with maximal q(G) in G(m,g) and H(m,c), respectively.

Extremal results on distance Laplacian spectral radius of graphs

The distance Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. We determine the unique trees with maximum distance Laplacian spectral radius among trees of perfect matching with given maximum degree, the unique trees with second (third, respectively) maximum distance Laplacian spectral radius, and the unique bipartite unicyclic graphs with maximum distance Laplacian spectral radius. We also determine the unique graphs with minimum distance Laplacian spectral radius among bicyclic graphs.

Lower bounds for the Laplacian spectral radius of graphs

Let G=(V,E) be a graph of order n and with the Laplacian spectral radius λ(G). For vi∈V, denote the set of all neighbors of vi by Ni and its number by di. The maximum degree of G is denoted by Δ(G). It is shown that if G is connected and Δ(G)<n−1 thenλ(G)≥max⁡{mi′+(1+(mi′−1)2d2,i)dimi′:vi∈V}, where mi′=∑vivj∈E(dj−|Ni∩Nj|)di and d2,i is the number of vertices at distance two from vi.Also it is shown thatλ(G)≥max⁡{(pij+(1−pij)2pijmax⁡{1,di−1})×|Ni∪Nj|:vivj∈E,di≥dj}, where pij=e(Ni,Nj−Ni)di(|Ni∪Nj|−di), e(Ni,Nj−Ni) is the number of edges between Ni and Nj−Ni.

The maximal A α-spectral radius of graphs with given matching number

Let be the set of simple graphs of order n with given matching number β. In this paper, we characterize the extremal graphs with maximal -spectral radius in for , which generalizes the results for adjacency matrix in [Feng L, Yu G, Zhang X-D. Spectral radius of graphs with given matching number. Linear Algebra Appl. 2007;422:133–138, Theorem 1.1] and signless Laplacian matrix in [Yu G. On the maximal signless Laplacian spectral radius of graphs. Proc Japan Acad Ser A. 2018, Theorem 2.5].

On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.

The signless Laplacian spectral radius of graphs with no intersecting triangles

Let Fk denote the k-fan consisting of k triangles which intersect in exactly one common vertex, and Sn,k the complete split graph of order n consisting of a clique on k vertices and an independent set on the remaining vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, it is shown that Sn,k is the unique graph attaining the maximum signless Laplacian spectral radius among all graphs of order n containing no Fk, provided that k≥2 and n≥3k2−k−2.

The signless Laplacian spectral radius of graphs with a prescribed number of edges

The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian matrix. In the paper, the graph with maximal signless Laplacian spectral radius among all graphs with given size and clique number is characterized. As applications, we determine the graph with maximal signless Laplacian spectral radius among all graphs with given size, and characterize the graph with maximal signless Laplacian spectral radius among all graphs with given size and chromatic number.

The signless Laplacian spectral radius of graphs with forbidding linear forests

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k⋅P3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k⋅P3 as a subgraph, respectively.

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G) = {v1, v2,…, vn}. The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n × n diagonal matrix with its (i, i)-entry equal to TrG(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as $$\mathscr{Q}(G)=\operatorname{Tr}(G)+D(G)$$ , where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.

Sharp upper bounds for the adjacency and the signless Laplacian spectral radius of graphs

Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has at least one common adjacent vertex. Our results improve some known upper bounds. The main tool we use here is the Lagrange identity.

The Randić index and signless Laplacian spectral radius of graphs

Given a connected graph G, the Randić index R(G) is the sum of (d(u)d(v))−1∕2 over all edges {u,v} of G, where d(u) and d(v) are the degrees of vertices u and v respectively. Let q(G) be the largest eigenvalue of the signless Laplacian matrix of G and n=|V(G)|. Hansen and Lucas (2010) made the following conjecture: q(G)R(G)≤4n−4n4≤n≤12;nn−1n≥13,where the equality holds if and only if G=Kn for 4≤n≤12 and G=Sn for n≥13, respectively. Here Kn is the complete graph with n vertices and Sn is the star with n vertices. Deng et al. verified this conjecture for 4≤n≤11. In this paper, we prove the conjecture for n≥12.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

Proof of a conjecture on the distance Laplacian spectral radius of graphs

Let G be a connected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G). The distance Laplacian matrix of G is defined as DL(G)=Tr(G)−D(G), where D(G) is the distance matrix and Tr(G)=diag(trv1,trv2,…,trvn) is the diagonal matrix of vertex transmissions of G. The largest eigenvalue of DL(G) is called the distance Laplacian spectral radius of G. In this paper, we obtain a graft transformation of a connected graph, which increases its distance Laplacian spectral radius. Using this transformation, we prove a conjecture involving the distance Laplacian spectral radius.

Bounds for the Laplacian spectral radius of graphs

This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds are given as functions of graph parameters like the number of vertices, the number of edges, degree sequence, average 2-degrees, diameter, covering number, domination number, independence number and other parameters.