Spectral Radius Of Graphs Research Articles

L-connectivity, l-edge-connectivity and spectral radius of graphs

l-connectivity, l-edge-connectivity and spectral radius of graphs

The Graft Transformation and Their Application on the Spectral Radius of Block Graphs

Let [Formula: see text] be a connected graph and [Formula: see text] be the adjacency matrix of [Formula: see text]. Suppose that [Formula: see text] are the eigenvalues of [Formula: see text]. In this paper, we first give a graft transformation on the spectral radius of graphs and then as their application, we determine the extremal graphs with maximum and minimum spectral radii among all clique trees. Furthermore, we also determine the unique graph with maximum spectral radius among all block graphs by using different methods.

On the spectral radius of graphs with given maximum degree and girth

We prove upper bounds for the spectral radius ρ(G) of an n-vertex graph with given maximum degree and girth at least 2ℓ+1. This extends the previous result of [9] regarding graphs with girth at least five. When ℓ=3 or |V(G)| is relatively small compared with the maximum degree, our upper bounds are sharp. In addition, for a tree T, we provide an upper bound for the spectral radius of an n-vertex T-free graph with given maximum degree. This bound is also sharp for a certain class of trees.

Spectral Radius of Graphs with Given Size and Odd Girth

Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m,k)>\sqrt{m-k+3}$ such that every non bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m,k)$ must contain an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng [The Electronic J. Combin. 29 (4) (2022)] as well.

Maximum degree and spectral radius of graphs in terms of size

Maximum degree and spectral radius of graphs in terms of size

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

The Extended Adjacency Indices for Several Types of Graph Operations

Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index <I>EA(G)</I> is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.

On a conjecture of spectral extremal problems

For a simple graph F, let Ex(n,F) and Exsp(n,F) denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an n-vertex graph without any copy of the graph F, respectively. The Turán graph Tn,r is the complete r-partite graph on n vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let F be any graph such that the graphs in Ex(n,F) are Turán graphs plus O(1) edges. Then Exsp(n,F)⊂Ex(n,F) for sufficiently large n. In this paper we consider the graph F such that the graphs in Ex(n,F) are obtained from Tn,r by adding O(1) edges, and prove that if G has the maximum spectral radius among all n-vertex graphs not containing F, then G is a member of Ex(n,F) for n large enough. Then Cioabă, Desai and Tait's conjecture is completely solved. Furthermore, we give a stronger result.

On the $$A_\alpha $$-spectral Radius of Graphs Without Large Matchings

On the $$A_\alpha $$-spectral Radius of Graphs Without Large Matchings

Maxima of the Q-index: Graphs with no K1,t-minor

A graph is said to be H-minor free if it does not contain H as a minor. In spectral extremal graph theory, it is interesting to determine the maximum (signless Laplacian) spectral radius of graphs that do not contain a given H as a minor. In this paper, we characterize the unique extremal graph with the maximum signless Laplacian spectral radius among all n-vertex connected K1,t-minor free graphs (t≥3).

The spectral radius of graphs with no intersecting odd cycles

Let Hs,t1,…,tk be the graph with s triangles and k odd cycles of lengths t1,…,tk≥5 intersecting in exactly one common vertex. Recently, Hou et al. (2018) [27], and Yuan (2018) [42] determined independently the maximum number of edges in an n-vertex graph that does not contain Hs,t1,…,tk as a subgraph. In this paper, we determine the graphs of order n that attain the maximum spectral radius among all graphs containing no Hs,t1,…,tk for n large enough.

On the Zero Forcing Number and Spectral Radius of Graphs

In this paper, we determine the graphs (respectively, trees) with maximum spectral radius among all graphs (respectively, trees) with zero forcing number at most $k$. As an application, we give a sharp lower bound for the zero forcing number of graphs involving the spectral radius.

The spanning k-trees, perfect matchings and spectral radius of graphs

A k-tree is a spanning tree in which every vertex has degree at most k. In this paper, we provide a sufficient condition for the existence of a k-tree in a connected graph with fixed order in terms of the adjacency spectral radius and the signless Laplacian spectral radius, respectively. Also, we give a similar condition for the existence of a perfect matching in a balanced bipartite graph with fixed order and minimum degree.

The spectral radius of graphs with no odd wheels

The odd wheel W2k+1 is the graph formed by joining a vertex to a cycle of length 2k. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W2k+1. We determine the structure of the spectral extremal graphs for all k≥2,k⁄∈{4,5}. When k=2, we show that these spectral extremal graphs are among the Turán-extremal graphs on n vertices that do not contain W2k+1 and have the maximum number of edges, but when k≥9, we show that the family of spectral extremal graphs and the family of Turán-extremal graphs are disjoint.

Measure-theoretic bounds on the spectral radius of graphs from walks

Let G be an undirected graph with adjacency matrix A and spectral radius ρ. Let wk,ϕk and ϕk(i) be, respectively, the number walks of length k, closed walks of length k and closed walks starting and ending at vertex i after k steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that wk,ϕk and ϕk(i) can be interpreted as the moments of three different measures, all of them supported on the spectrum of A. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on ρ, as well as provide alternative proofs to several well-known bounds in the literature.

On the spectral radius of graphs without a star forest

Let F=∪i=1kSdi be the union of pairwise vertex-disjoint k stars of order d1+1,…,dk+1, respectively, where k≥2 and d1≥⋯≥dk≥1. In this paper, we present two sharp upper bounds for the spectral radius of F-free (bipartite) graphs and characterize all corresponding extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of an F-free graph and all extremal graphs are obtained.

Spectral extrema of graphs: Forbidden hexagon

To determine the Turán numbers of even cycles is a central problem of extremal graph theory. Even for C6, the Turán number is still open. Till now, the best known upper bound is given by Füredi, Naor and Verstraëte [On the Turán number for the hexagon, Advances in Math.]. In 2010, Nikiforov posed a spectral version of extremal graph theory problem: what is the maximum spectral radius ρ of an H-free graph of order n? Let exsp(n,H)=max{ρ(G)||V(G)|=n,H⊈G}. In contrast to the unsolved problem of Turán number of C6, we obtain the exact value of exsp(n,C6) and characterize the unique extremal graph. The result also confirms Nikiforov’s conjecture [The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl.] for k=2.

A note on the ABC spectral radius of graphs

ABSTRACT The ABC matrix of a graph G, recently introduced by Estrada, is the square matrix of order whose -entry is if the ith vertex and the jth vertex of G are adjacent, and 0 otherwise, where is the degree of the ith vertex of G. The ABC spectral radius of G is the largest eigenvalue of the ABC matrix of G. In this paper, we completely characterize the (connected) graphs and (connected) triangle-free graphs which have the maximum, the minimum and second-minimum ABC spectral radius. These results answer a general question posed in [X. Chen, On ABC eigenvalues and ABC energy, Linear Algebra Appl. 544 (2018) 141–157] for (connected) graphs and (connected) triangle-free graphs.

A bound on the spectral radius of graphs in terms of their Zagreb indices

The first and the second Zagreb index of a graph, usually defined as the sum of the squares of degrees over all vertices and the sum of the products of degrees of edge endvertices over all edges, respectively, are tightly related to the numbers of walks of length two and three in the graph. We provide here a lower bound on the spectral radius of adjacency matrix A of graph in terms of its Zagreb indices, based on the properties of the least square approximation of the vector A2j with the vectors Aj and j, where j is the all-one vector. The bound is sharp for all graphs with two main eigenvalues, surpassing the range of sharpness of other bounds among connected graphs.

A conjecture on the spectral radius of graphs

Let G be a simple graph of order n and also let ρ(G) be the spectral radius of the graph G. In this paper, we prove that ρ(G)≤ρ2(G−vk)+2dk−1 for any non-isolated vertex vk of degree dk in G, thus confirming a conjecture posed by Guo et al. (2019) [3]. Moreover, we characterized all connected graphs for which this bound is attained.