The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the maximum $\varepsilon$-spectral radius of trees with given order. In this paper, we consider the above problem of $n$-vertex trees with given diameter. The maximum $\varepsilon$-spectral radius of $n$-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. The trees with least $\varepsilon$-eigenvalues in $[-2\sqrt{2},0)$ have been known. Finally, we determine the trees with least $\varepsilon$-eigenvalues in $[-2-\sqrt{13},-2\sqrt{2})$.
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. This matrix is first defined in 2018 by Wang et al. \cite{1}. In this paper we have characterized the graphs which have at most one (hence exactly) positive eigenvalue of $\varepsilon(G)$.
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star.…
The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the s…
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and cha…
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G),$ and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of $\varepsilon$-eigenvalues. In this paper, we establish some bounds for the $\varepsilon$-energy of the complete multipartite graph $K_{n_1, n_2, \ldots,n_p}$ of order $n= \sum_{i=1}^p n_i $ and characterize the extreme graphs. …
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of $\varepsilon(G)$ is called the $\varepsilon$-spectral radius, and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\varepsilon(G)$. We establish some bounds for the $\varepsilon$-spectr…
The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks have at most two cut-vertices \textcolor{blue}{of the clique tree}. After proving the irreducibility of the eccentricity matrix of a clique tree in $\mathcal C \mathcal T$ and finding its inertia indices, we show that every graph in $\mathcal C \mathcal T$ with more than $4$ vert…
In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index τ(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.
We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq λ_N\leq 2$, to the case of chemical hypergraphs.
The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of all $n$-vertex trees, $\mathbb{T}_{n,Δ}$ be the set of $n$-vertex trees with given maximum degree equal to $Δ$, $\mathbb{T}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree, and let $\mathbb{T}_{n,Δ}^k$ be the set of $n$-vertex trees with exactly $k$ vertices of given maximum degree equal to $Δ.$ In this paper, we first determine…
In this article, we establish some bounds involving the largest two distance Pareto eigenvalues of a connected graph. Also we characterize all possible values for smallest six distance Pareto eigenvalues of a connected graph.
We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graph with smallest eigenvalue $-1-\frac{b_1}{2}$.
We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem.
The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.
We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.
Let $v(k, λ)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $λ$. The Alon-Boppana Theorem implies that $v(k, λ)$ is finite when $k > \frac{λ^2 + 4}{4}$. In this paper, we show that for fixed $λ\geq1$, there exists a constant $C(λ)$ such that $2k+2 \leq v(k, λ) \leq 2k + C(λ)$ when $k > \frac{λ^2 + 4}{4}$.
For every real $0\leq α\leq 1$, Nikiforov defined the $A_α$-matrix of a graph $G$ as $A_α(G)=αD(G)+(1-α)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. The eigenvalues of $A_α(G)$ are called the $A_α$-eigenvalues of $G$. Let $S_k(A_α(G))$ be the sum of $k$ largest $A_α$-eigenvalues of $G$. In this paper, we present several upper and lower bounds on $S_k(A_α(G))$ and characterize the extremal graphs for certain cases, which can be regard as a common generalization of the sum of $k$ largest eigenvalues of adjacency matrix and s…
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.
The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and average degree of the vertices of G. Motivated by the work from Sharafdini et al. [R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and other topological indices. J. Math. Nanosci. 2016, 6, 49-57.], in this paper we investigate the eccentricity version of Laplacian energy of a graph G.
The flip-graph of a convex polygon $π$ is the graph whose vertices are the triangulations of $π$ and whose edges correspond to flips between them. The eccentricity of a triangulation $T$ of $π$ is the largest possible distance in this graph from $T$ to any triangulation of $π$. It is well known that, when all $n-3$ interior edges of $T$ are incident to the same vertex, the eccentricity of $T$ in the flip-graph of $π$ is exactly $n-3$, where $n$ denotes the number of vertices of $π$. Here, this statement is generalized to arbitrary triangulations. Denoting by $n-3-k$ the largest number of inter…
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of $G$ is sum of the absolute values of the eigenvalues of $\varepsilon(G)$. Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such…
Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $σ$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^σ$. In this paper we prove that there exists a partial orientation $σ$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^σ$ is at most the largest absolute value of the roots of the matching polynomial of $G$.
Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$. Let $λ_{k}(G)$ be the $k$-th smallest normalized Laplacian eigenvalue of $G$. Denote $ρ(A)$ the spectral radius of the matrix $A$. In this paper, we study the behaviors of $λ_{2}(G)$ and $ρ(\mathcal{L}(G))$ when the graph is perturbed by three operations.
In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2Δ(G)$, where $Δ(G)$ is the largest degree of $G$, if and only if $G$ is regular. Thus the largest Laplacian H-eigenvalue of $G$, reaches the same upper bound, if and only if $G$ is regular and odd-bipartite. We show that an $s$-cycle $G$, as a $k$-uniform hypergraph, where $1 \le s \le k-1$, is regular if and only if there is a positive integer $q$ such that $k=q(k-s)$. We show that an even-uniform $s$-path and an even-uniform non-regular…
We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when $d$ is sufficiently large. Our proofs rely on the method of compressions.
Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $λ_{1}(D(G))\geqλ_{2}(D(G))\geq\cdots\geqλ_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral characterization on the second largest distance eigenvalue $λ_{2}(D(G))$ of graphs, and prove that the graphs with $λ_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by their $D$-spectra.
In this article, we study Pareto eigenvalues of distance matrix of connected graphs and show that the non zero entries of every distance Pareto eigenvector of a tree forms a strictly convex function on the forest generated by the vertices corresponding to the non zero entries of the vector. Besides we find the minimum number of possible distance Pareto eigenvalue of a connected graph and establish lower bounds for $n$ largest distance Pareto eigenvalues of a connected graph of order $n.$ Finally, we discuss some bounds for the second largest distance Pareto eigenvalue and find graphs with opti…
Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1} then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding theory and cryptography. As many of these applications depend on the size of the largest non-trivial po…
We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $λ_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$), we have $$ M(G) \le λ_{\max}(G) \le \big(\frac14 \log|V| + 1 \big) \M(G). $$ Here the lower bound is attained if $G$ is regular or bi-regular, whereas the logarithmic factor in the upper bound, conjecturally, can be improved --- although we present an example showing that it cannot be replaced with a factor growing slower than $(\log |V|/\log\log|V|)^{1/8}$. Furthe…
Let $A_n$ denote the alternating group of degree $n$ with $n\geq 3$. The alternating group graph $AG_n$, extended alternating group graph $EAG_n$ and complete alternating group graph $CAG_n$ are the Cayley graphs $\mathrm{Cay}(A_n,T_1)$, $\mathrm{Cay}(A_n,T_2)$ and $\mathrm{Cay}(A_n,T_3)$, respectively, where $T_1=\{(1,2,i),(1,i,2)\mid 3\leq i\leq n\}$, $T_2=\{(1,i,j),(1,j,i)\mid 2\leq i<j\leq n\}$ and $T_3=\{(i,j,k),(i,k,j)\mid 1\leq i<j<k\leq n\}$. In this paper, we determine the second largest eigenvalues of $AG_n$, $EAG_n$ and $CAG_n$.
The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type result on $W(G)$ involving $\varepsilon(G)$ is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $W(T) - \varepsilon(T)$ is minimized on caterpillars. An exact formula for $W(T) - \varepsilon(T)$ in terms of the radius of a tree $T$ is obt…
Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrix eigenvalues $μ_1(G)\geq μ_2(G)\geq \dots \geq μ_n(G)$. In this paper we study the quantity $$μ_1(G)+μ_n(G).$$ We prove that for any triangle-free graph $G$ we have $$μ_1(G)+μ_n(G)\leq (3-2\sqrt{2})n.$$ This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum of $$\frac{μ_1(G)+μ_n(G)}{n}.$$
Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications
School of Science, Zhejiang University of Science and Technology（浙江科技大学理学院）；School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；Department of Mathematics, Sungkyunkwan University（成均馆大学数学系）
For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=λ_1(D(G))+\cdots+λ_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and lower bound of $S_k(D(G))$. We determine the sharp lower bounds of $S_k(D(G))$ when $G$ is connected graph and is a tree, respectively, and characterize both the extremal graphs. Moreover, we conjecture that the upper bound is attained when $G$ is a path of order $n$ and prove some partial result supporting the conjecture. To prove our result, we obtain a sharp upper boun…
We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.
The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article [Jianfeng Wang, Mei Lu, Francesco Belardo, Milan Randic. The anti-adjacency matrix of a graph: Eccentricity matrix. Discrete Applied Mathematics, 251: 299-309, 2018.], is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classe…
We prove that for all values of the edge probability p(n) the largest eigenvalue of a random graph G(n,p) satisfies almost surely: λ_1(G)=(1+o(1))max{\sqrtΔ,np}, where Δis a maximal degree of G, and the o(1) term tends to zero as max{\sqrtΔ,np} tends to infinity.
