Bounds for Incidence Energy of Some Graphs

Abstract

Let be a simple graph. The incidence energy ( for short) of is defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound for of graphs in terms of the maximum degree is given. Meanwhile, bounds for of the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.

Some improved bounds on two energy-like invariants of some derived graphs

“…, qn} is the Q-spectrum of G. Notice that q = r as G is r-regular. Then from(2) and the (ii) in Lemma 2.2, it is easy to see that, by a simple calculation,IE(Q(G)) = n i= r + q i − + r( r + q i − ) − q i + (m − n) i= q i = m − r = (n − )r.Applying the Cauchy-Schwarz inequality, one obtainsn i= r + q i − + r( r − + q i ) − q i ≤ (n − ) n i= ( r + q i − + r( r − + q i ) − q i ) − + q i ) − q i ≤ (n − ) n− n− r − + n− (n − ) n i= [r( r − + q i ) − q i ] = (n − ) n− n− r − + n− n− r(r − ),which, along with(24), implies the desired upper bound. Moreover, above equality occurs if and only if q = r and q = q = • • • = qn.…”

“…Recall that the line graph L(G) [2] of G is the graph whose vertex set is the edge set of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have exactly a common vertex. Given an (r , r )-semiregular graph G of order n with m edges, then the L-spectrum [23] and Q-spectrum [25]…”

“…In addition, Gutman et al [21] presented some sharp bounds for IE of the line graph and iterated line graph of regular graphs. Wang et al [24] gave some new bounds for IE of the subdivision graph and total graph of regular graphs. Wang and Yang [25] also presented some upper bounds on IE for the line graph of semiregular graphs, the para-line graph of a regular graph.…”

“…Wang et al [24] gave some new bounds for IE of the subdivision graph and total graph of regular graphs. Wang and Yang [25] also presented some upper bounds on IE for the line graph of semiregular graphs, the para-line graph of a regular graph. Recently, Chen et al [26] obtained some new bounds for LEL and IE of the line graph, subdivision graph and total graph of regular graphs.…”

The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

“…For the graphs with the extremal IEs and the upper and lower bounds of IE, one can refer to Refs. [5,7,10,15,20,21,24,26,27]. Kaya and Maden got some bounds for the generalized version of incidence energy [12].…”