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## Abstract

Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has 1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of order n with k negative edges, k ---lt--- n − 1 and has maximum index, then negative edges form K 1 ,k. In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities:

−5 ≤ λn ≤ ≤ λ2 ≤ 3.

## Abstract

Slobodan Simić1 had many interests and many friends. Doubtless each of his 66 co-authors2 has a story to tell, but here we can offer only our own personal perspectives.

## Abstract

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.

## Abstract

A regular graph is co-edge regular if there exists a constant µ such that any two distinct and non-adjacent vertices have exactly µ common neighbors. In this paper, we show that for integers s ≥ 2 and n large enough, any co-edge-regular graph which is cospectral with the s-clique extension of the triangular graph T (n) is exactly the s-clique extension of the triangular graph T (n).

## Abstract

Let G be a graph with m edges and let ρ be the largest eigenvalue of its adjacency matrix. It is shown that

$ρ≤2(1-⌊1/2+2m+1/4⌋-1)m,$

improving the well-known bound of Stanley. Moreover, writing ω for the clique number of G and Wk for the number of its walks on k vertices, it is shown that the sequence

${((1-1/ω)W2k)1/2k}k=1∞$

is nonincreasing and converges to ρ.

## Abstract

If in the right-hand sides of given differential equations occur discontinuities in the state variables, then the natural notion of a solution is the one in the sense of Filippov. In our paper, we will consider this type of solutions for vector Dirichlet problems. The obtained theorems deal with the existence and localization of Filippov solutions, under effective growth restrictions. Two illustrative examples are supplied.

## Abstract

We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over ℚ(m). It is given by the equation

$Em:y2=x3−3636m4+48m2+2536m4−48m2+25x.$

Employing canonical heights we show the rank is in fact at least three for all m. Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve Em are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [2, 3]).

## Abstract

If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.