Graphs with Given Degree Sequence and Maximal Spectral Radius
Tuerker Biyikoglu, Josef Leydold
备注：
12 pages, 4 figures; revised version. Important change: Theorem 3 (formely Theorem 7) now states (and correctly proofs) the majorization result only for "degree sequences of trees" (instead for general connected graphs). Bo Zhou from the South China Normal University in Guangzhou, P.R. China, has found a counter-example to the stronger result
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 show that the kite graph $K_4^{(n)}$ uniquely maximizes the distance spectral radius among all connected $4$-chromatic planar graphs on $n$ vertices.
Let $\mathscr{G}_{n,β}$ be the set of graphs of order $n$ with given matching number $β$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_α(G)=αD(G)+A(G)$ is called the $α$-spectral radius of $G$. The graphs with maximal $α$-spectral radius in $\mathscr{G}_{n,β}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $\mathscr{G}_{n,β}$. More precisely, we generalize the known results on t…
We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through elementary methods.
We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.
This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and quasi-radius. Moreover, maximal digraphs with finite values of radius and quasi-diameter are studied.
Let $\mathcal{U}(n,k)$ and $Γ(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of transformations and constructing the incidence matrices for the hypergraphs considered, we get the hypergraphs with the maximal spectral radii among three kinds of hypergraphs, namely $\mathcal{U}(n,k)$ with $n= 2k(k-1)$ and $n\geq 9k(k-1)$, $Γ(n,k)$ with $n\geq k(k-1)$, and $\mathcal{U}(n,k)\cup Γ(n,k)$ with $n\geq 2k(k-1)$, where $k\geq 3$.
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. We apply the various bounds to the classes of maximal rank distance and quasi maximal rank distance codes.
A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order $n$. Furthermore, we determine the first three unicyclic, and bicyclic, $C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.
The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n "lie close to each other" in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n < 221. The results are consistent with a recent conjecture on the limiting shape of the strict partition of maximal projective degree.
We consider the problem of computing 2-center in maximal outerplanar graph. In this problem, we want to find an optimal solution where two centers cover all the vertices with the smallest radius. We provide the following result. We can compute the optimal centers and the optimal radius in $O(n^2)$ time for a given maximal outerplanar graph with $n$ vertices. We try to let the maximal outerplanar graph be cut into two subgraphs with an internal edge, each center will cover vertices which are continuous.
We prove that every graph of minimum degree at least $d \ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelikán ($d=4$). We conjecture that for any planar maximal 3-degenerate graph $H$ of order $d+1$ and any graph $G$ of minimum degree at least $d$, $G$ contains a subdivision of $H$. We verify this in the case $H$ is $P_6^3$ and $P_7^3$
In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.
Collatz and Sinogowitz had proposed to measure the departure of a graph $G$ from regularity by the difference of the (adjacency) spectral radius and the average degree: $ε(G)=ρ(G)-\frac{2m}{n}$. We give here new lower bounds on this quantity, which improve upon the currently known ones.
It is known that the spectral radius of a digraph with k edges is \le \sqrt{k}, and that this inequality is strict except when k is a perfect square. For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, and a corresponding loop, and then connecting it to the first \lfloor \ell/2\rfloor vertices by pairs of directed edges (this is for odd \ell, for even \ell we add one extra edge to the new vertex). Using a combinatorial reciprocity theorem by Gessel, and a classification by Backelin on t…
The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.
We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of Stevanovic, Zhang, and Alon and Sudakov.
Let $G$ be an irregular graph on $n$ vertices with maximum degree $Δ$ and diameter $D$. We show that Δ-λ_1>\frac{1}{nD} where $λ_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.
Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $κ(G) \leq k$ and showed that among those graphs, the maximum spectral radius is obtained uniquely at $K_k^n$, which is the graph obtained by joining $k$ edges from $k$ vertices of $K_{n-1}$ to an isolated vertex. In this paper, we study the spectral radius of graphs of order $n$ with $κ(G)\leq k$ and minimum degree $δ(G)\geq k $. We show that among those graphs, the maximum spectral radius is obtained uniquely at $K_{…
Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp upper bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.
Let $A_α(G)$ be the $A_α$-matrix of a digraph $G$ and $λ_{α1}, λ_{α2}, \ldots, λ_{αn}$ be the eigenvalues of $A_α(G)$. Let $ρ_α(G)$ be the $A_α$ spectral radius of $G$ and $E_α(G)=\sum_{i=1}^n λ_{αi}^2$ be the $A_α$ energy of $G$ by using second spectral moment. Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_α$ spectral radius and the maximal (minimal) $A_α$ …
Let $H_n$ be the cactus obtained from the star $K_{1,n-1}$ by adding $\lfloor \frac{n-1}{2}\rfloor$ independent edges between pairs of pendant vertices. Let $K_{1,n-1}^+$ be the unicyclic graph obtained from the star $K_{1,n-1}$ by appending one edge. In this paper we give alternative proofs of the following results: Among all cacti with $n$ vertices, $H_n$ is the unique cactus whose spectral radius is maximal, and among all unicyclic graphs with $n$ vertices, $K_{1,n-1}^+$ is the unique unicyclic graph whose spectral radius is maximal. We also prove that among all odd-cycle graphs with $n$ ve…
In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.
The general spectral radius of a graph $G$, denoted by $Θ(G,α)$, is the maximal eigenvalue of $M_α(G)=A(G)+αD(G)$ $(α\geq 0)$, 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 $Θ_α$-maximal in a class of connected simple graphs $\mathcal {G}$ if $Θ(G,α)$ 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 $π=\big(d_1,d_2,\ldots,d_n\big)$ and $π'=\big(d'_1,d'_2,\ldots,d'_n\big)$ be two non-increasing degre…
Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a …
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 $α\in[0,1]$, write $A_α(G)$ for the matrix $$A_α(G)=αD(G)+(1-α)A(G).$$ This paper presents some extremal results about the spectral radius $ρ(A_α(G))$ of $A_α(G)$ that generalize previous results about $ρ(A_0(G))$ and $ρ(A_{\frac{1}{2}}(G))$. In this paper, we give some results on graph perturbation for $A_α$-matrix with $α\in [0,1)$. As applications, we characterize all extremal trees with the maximum $A_α$-spectral radius in the set of all trees with prescribed degree se…
Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.
For a tree with the given sequence of vertex degrees the spectral radius of its terminal distance matrix is shown to be bounded from below by the the average row sum of the terminal distance matrix of the, so called, BFS-tree (also known as a greedy tree). This lower bound is typically not tight; nevertheless, since spectral radius of the terminal distance matrix of BFS-tree is a natural upper bound, the numeric simulation shows that relative gap between the upper and the lower bound does not exceed 3% (we also make a step towards justifying this fact analytically.) Therefore, the conjecture t…
A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{π^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum…
To study how balanced or unbalanced a maximal intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is we consider the ratio $\mathcal{R}(\mathcal{F})=\frac{Δ(\mathcal{F})}{δ(\mathcal{F})}$ of its maximum and minimum degree. We determine the order of magnitude of the function $m(n,r)$, the minimum possible value of $\mathcal{R}(\mathcal{F})$, and establish some lower and upper bounds on the function $M(n,r)$, the maximum possible value of $\mathcal{R}(\mathcal{F})$. To obtain constructions that show the bounds on $m(n,r)$ we use a theorem of Blokhuis on the minimum size of a non-trivial bl…
The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of $\mathrm{PGL}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.
In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices, its maximum degree is at least $\frac 12(1-\frac {k^2}n) m - {n-2\choose k-2}\ge 0.49 m$. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on non-empty subgraphs with small maximum degree.
Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $ρ(G)$ and $q(G)$ the spectral radius and the signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge 0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. We show that if $ρ(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or $K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then $K_{1,m-k}\subseteq G$. Both these two results extend some known results.
In 1969 Denniston gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of isomorphism classes of Mathon maximal arcs of degree 8 in PG(2,2^h), h prime.
A path of a graph $G$ is called a Hamilton path if it passes through all the vertices of $G$. A graph is Hamilton-connected if any two vertices are connected by a Hamilton path. Note that any bipartite graph is not Hamilton-connected. We consider the weak version of Hamilton-connected property among bipartite graphs. A weakly Hamilton-connected graph is a balanced bipartite graph $G=(X,Y,E)$ that there is a Hamilton path between any vertex $x\in X$ and $y\in Y$. In this paper, we present some degrees, number of edges, and spectral radius conditions for a simple balanced bipartite graph to be w…
For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $Δ\geq 3$ and $t\geq 3$, denote by $T_a(Δ,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $Δ-1$ $P_2$'s on each end of the path $P_t$, and $T_b(Δ, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $Δ-1$ $P_2$'s on an end of the $P_{t+2}$ and $Δ-2$ $P_2$'s on the vertex next to the end. In [X. Li, X. Yao, J. Zhang and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math. Chem. 45(2009), 962…
Given a connected graph $G$ on $n$ vertices, with minimum degree $δ\geq 2$ and girth at least $g \geq 4$, what is the maximum radius $r$ this graph can have? Erdős, Pach, Pollack and Tuza established in the triangle-free case ($g=4$) that $r \leq \frac{n-2}δ+12$, and noted that up to the value of the additive constant, this is tight. We determine the exact value for the triangle-free case. For higher $g$ little is known. We settle the order of $r$ for $g=6,8,12$ and prove an upper bound to the order for general even $g$. Finally, we show that proving the corresponding lower bound for general e…
It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension $(4s-2)$ with degree $s$ which are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as "Cayley conjecture". Moreover, by modifying Nill's inequaitly sli…
Let $F$ be a planar, $3$-connected graph of radius one on $p$ vertices, with $a$ vertices of degree three. We characterise all unigraphic degree sequences for such graphs, when $a\geq 3$ and $p$ is large enough with respect to $a$. This complements the work of \cite{mafpo4} for $a\leq 3$.
In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is $(\floor {m/2} +1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $…
For $0\le α\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_α(G)=αD(G)+(1-α)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $α$-spectral radius of $G$ is the largest eigenvalue of $A_α(G)$. We give upper bounds for $α$-spectral radius for unicyclic graphs $G$ with maximum degree $Δ\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $α$-spectral radius…
To estimate the lower bound for the chromatic number of a graph $G$, Lovász associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent ei…
The existence of a spanning tree with leaf distance at least $d$ and leaf degree at most $k$ via the size or the spectral radius with respect to the minimum degree
For $n\ge 6$ let $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2}$, $v_1v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},v_2v_{n-1}\}$, $E_3=\{v_0v_1,\ldots,v_0v_{n-4}$, $v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$, $T_n^3=(V,E_1),\ T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3).$ In this paper, for $p\ge n\ge 15$ we obtain explicit formulas for $ex(p;T_n^3)$, $ex(p;T_n^{''})$ and $ex(p;T_n^{'''})$, where $ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph.
Let $\mathcal{G}_{n, β^*}$ $(\mathcal{G}^*_{n,β^*})$ be the set of all (connected) graphs of order $n$ with fractional matching number $β^*$. In this paper, the graphs with maximal spectral radius in $\mathcal{G}_{n,β^*}$ and $\mathcal{G}^*_{n,β^*}$ are characterized, respectively. Moreover, a lower bound for the spectral radius in graphs with order $n$ to guarantee the existence of a perfect fractional matching is also given, which generalizes the main result of O [Suil O, Spectral radius and matchings in graphs, Linear Algebra and its Applications, 2020].
Let $G$ be a connected graph with minimum degree $δ(G)$ and vertex-connectivity $κ(G)$. The graph $G$ is $k$-connected if $κ(G)\geq k$, maximally connected if $κ(G) = δ(G)$, and super-connected (or super-$κ$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough.
Given a permutation P in Sn, let G(P) be the graph on n vertices {1,...,n}, where two vertices i<j are adjacent if i appears right of j in P and there are no integers k with i<k<j and k appearing between i and j in P. Let G'(P) be the graph obtained by dropping the condition that i appears right of j, i.e. two vertices are adjacent if the rectangle [i,P(i)] x [j,P(j)] is empty. In the study of the strong order on permutation, Adin and Roichman introduced these graphs and computed their maximum number of edges. We generalize these results to the Weyl group of signed permutations Bn, working wit…
We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$. We also find all $3$-polytopes of radius one with $p\leq 17$, and those with $q\leq 41$ edges, by developing a fast algorithm and making use of High Performance Computing.
N. Hamilton and J. A. Thas describe a link between maximal arcs of Mathon type and partial flocks of the quadratic cone. This link is of a rather algebraic nature. In this paper we establish a geometric connection between these two structures. We also define a composition on the flock planes and use this to work out an analogue of the synthetic version of Mathon's Theorem. Finally, we show how it is possible to construct a maximal arc of Mathon type of degree 2d, containing a Denniston arc of degree d provided that there is a solution to a certain given system of trace conditions.
Let $G$ be the symmetric group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that both functions $σ(G)$ and $ω(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $ω(G)$ which is independent of the classification of finite…
