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On strongly regular graphs with m2 = qm3 and m3 = qm2

Lepovic, Mirko (2011)

Serdica Mathematical Journal

2010 Mathematics Subject Classification: 05C50.We say that a regular graph G of order n and degree r і 1 (which is not the complete graph) is strongly regular if there exist non-negative integers t and q such that |SiЗSj| = t for any two adjacent vertices i and j, and |SiЗSj| = q for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let l1 = r, l2 and l3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote...

On the bounds of Laplacian eigenvalues of k -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

Let μ n - 1 ( G ) be the algebraic connectivity, and let μ 1 ( G ) be the Laplacian spectral radius of a k -connected graph G with n vertices and m edges. In this paper, we prove that μ n - 1 ( G ) ≥ 2 n k 2 ( n ( n - 1 ) - 2 m ) ( n + k - 2 ) + 2 k 2 , with equality if and only if G is the complete graph K n or K n - e . Moreover, if G is non-regular, then μ 1 ( G ) < 2 Δ - 2 ( n Δ - 2 m ) k 2 2 ( n Δ - 2 m ) ( n 2 - 2 n + 2 k ) + n k 2 , where Δ stands for the maximum degree of G . Remark that in some cases, these two inequalities improve some previously known results.

On The Determinant of q-Distance Matrix of a Graph

Hong-Hai Li, Li Su, Jing Zhang (2014)

Discussiones Mathematicae Graph Theory

In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph...

On the distance spectrum of a cycle

Ante Graovac, Gani Jashari, Mate Strunje (1985)

Aplikace matematiky

Analytic expressions for the roots of the distance polynomial of a cycle are given.

On the inverse eigenvalue problem for a special kind of acyclic matrices

Mohammad Heydari, Seyed Abolfazl Shahzadeh Fazeli, Seyed Mehdi Karbassi (2019)

Applications of Mathematics

We study an inverse eigenvalue problem (IEP) of reconstructing a special kind of symmetric acyclic matrices whose graph is a generalized star graph. The problem involves the reconstruction of a matrix by the minimum and maximum eigenvalues of each of its leading principal submatrices. To solve the problem, we use the recurrence relation of characteristic polynomials among leading principal minors. The necessary and sufficient conditions for the solvability of the problem are derived. Finally, a...

On the inverse of the adjacency matrix of a graph

Alexander Farrugia, John Baptist Gauci, Irene Sciriha (2013)

Special Matrices

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary...

On the Laplacian energy of a graph

Mirjana Lazić (2006)

Czechoslovak Mathematical Journal

In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on n vertices ( n = 1 , 2 , ... ) . Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant α ≥ 4 , and completely describe this class in the case α = 40 .

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