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A Note on the Permanental Roots of Bipartite Graphs

Heping Zhang, Shunyi Liu, Wei Li (2014)

Discussiones Mathematicae Graph Theory

It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imaginary axes. Furthermore, we prove that any graph has no negative real permanental root, and any graph containing...

A note on tree realizations of matrices

Alain Hertz, Sacha Varone (2007)

RAIRO - Operations Research

It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + mkl ≤ max{mik + mjl, mil + mjk} for all distinct i,j,k,l.

A note on ultrametric matrices

Xiao-Dong Zhang (2004)

Czechoslovak Mathematical Journal

It is proved in this paper that special generalized ultrametric and special 𝒰 matrices are, in a sense, extremal matrices in the boundary of the set of generalized ultrametric and 𝒰 matrices, respectively. Moreover, we present a new class of inverse M -matrices which generalizes the class of 𝒰 matrices.

A result related to the largest eigenvalue of a tree

Gurusamy Rengasamy Vijayakumar (2008)

Discussiones Mathematicae Graph Theory

In this note we prove that {0,1,√2,√3,2} is the set of all real numbers l such that the following holds: every tree having an eigenvalue which is larger than l has a subtree whose largest eigenvalue is l.

A sharp upper bound for the spectral radius of a nonnegative matrix and applications

Lihua You, Yujie Shu, Xiao-Dong Zhang (2016)

Czechoslovak Mathematical Journal

We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.

A shorter proof of the distance energy of complete multipartite graphs

Wasin So (2017)

Special Matrices

Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

The imbalance of an edge e = { u , v } in a graph is defined as i ( e ) = | d ( u ) - d ( v ) | , where d ( · ) is the vertex degree. The irregularity I ( G ) of G is then defined as the sum of imbalances over all edges of G . This concept was introduced by Albertson who proved that I ( G ) ≤ 4 n 3 / 27 (where n = | V ( G ) | ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves the...

Algebraic conditions for t -tough graphs

Bo Lian Liu, Siyuan Chen (2010)

Czechoslovak Mathematical Journal

We give some algebraic conditions for t -tough graphs in terms of the Laplacian eigenvalues and adjacency eigenvalues of graphs.

Algebraic connectivity of k -connected graphs

Stephen J. Kirkland, Israel Rocha, Vilmar Trevisan (2015)

Czechoslovak Mathematical Journal

Let G be a k -connected graph with k ≥ 2 . A hinge is a subset of k vertices whose deletion from G yields a disconnected graph. We consider the algebraic connectivity and Fiedler vectors of such graphs, paying special attention to the signs of the entries in Fiedler vectors corresponding to vertices in a hinge, and to vertices in the connected components at a hinge. The results extend those in Fiedler’s papers Algebraic connectivity of graphs (1973), A property of eigenvectors of nonnegative symmetric...

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