Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues?

A survey on integral graphs

“…So far only some partial results concerning these graphs are known (some serious investigations are in progress). Basic results are contained in [5,6,7].…”

More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

“…For all other facts from the spectral graph theory (including integral graphs) the reader is referred to one of the books [4,6]. In this paper we solve the problem posed in [3]. The main result of this paper reads: Theorem 1.1.…”

“…, H * s * = H s+s * . Therefore, the following "graphical" representation of graphs from S arises (see ; an unoriented line stands if there are at least two edges between the subgraphs in question -note G is 2-connected (see [3] Proposition 2.1). Some important subgraphs of G ∈ S , in view of the above representation, are:…”

“…Namely, since q is a non-negative integer, 39 triplets are eliminated at once, which results in its reduced form, here addressed as 23 24:(1,18,4) 2: (3,2,12) 25: (3,14,6) 26: (5,10,8) 27:(7,6,10) 18 3:(2,2,14) 6: (2,10,6) 24 28:(4,10,10) 7: (4,6,8) 29:(6,6,12) 8: (6,2,10) 30:(8,2,14) 19 4:(1,2,16) 9: (1,10,8) 25 31:(5,6,14) 10: (3,6,10) 32: ( Proof. Expressing M 6 (G) in two ways, by definition, and by Lemma 2.1 (3 o ), we get 2(3 6 + 2 6 a) = 2m + 12f + 12g + 48q + 12e + 12h + 6p.…”

Some families of integral graphs

“…In general, the problem of characterizing integral graphs seems to be very difficult. Thus, it makes sense to restrict our investigations to some interesting families of graphs, for instance, cubic graphs [4,21], complete r -partite graphs [20,23], graphs with maximum degree 4 [2,3], 4-regular integral graphs [9], integral graphs which belong to the classes α K a,b or α K a ∪ β K b,b [16,17], etc. Trees represent another important family of graphs for which the problem has been considered in [1,5,6,[10][11][12][13][14][15]22,[24][25][26][27].…”