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Fig. 6. Two threshold graphs C’'S4,2 and C'S45 (Black vertices are dominating vertices).

Figure 6 Two threshold graphs C’'S4,2 and C'S45 (Black vertices are dominating vertices).

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We are now ready to give another upper bound on energy €(G) of graph G in terms of n,m, r, A and M,(G).
We are now ready to give another upper bound on energy €(G) of graph G in terms of n,m, r, A and M,(G).
The first part of the proof is done. and by the Cauchy-Schwarz inequality, we have
The first part of the proof is done. and by the Cauchy-Schwarz inequality, we have
Corollary 2.3.12. [72] Let G be a d-regular connected graph of order n. Ther Proof: Since r <n, the result follows from Theorem 2.3.10
Corollary 2.3.12. [72] Let G be a d-regular connected graph of order n. Ther Proof: Since r <n, the result follows from Theorem 2.3.10
2.4. Characterization of extremal bipartite graphs In the following theorem we characterize the extremal graphs for the above upper bound in (2.45). For this we need to give the same proof of the upper bound mentioned in [39]. In the following theorem we characterize the extremal graphs for the above
2.4. Characterization of extremal bipartite graphs In the following theorem we characterize the extremal graphs for the above upper bound in (2.45). For this we need to give the same proof of the upper bound mentioned in [39]. In the following theorem we characterize the extremal graphs for the above
From (2.47) with the above result, we get the required result in (2.46). The first part of the proof is done.
From (2.47) with the above result, we get the required result in (2.46). The first part of the proof is done.
Remark 3.2.3. Suppose that T (T # P,,) is any tree of order n. For T, we will From the proof of the Theorem 3.2.1, we get
Remark 3.2.3. Suppose that T (T # P,,) is any tree of order n. For T, we will From the proof of the Theorem 3.2.1, we get
Hence our result in (3.7) is better than the result in (3.6) for tree T.
Hence our result in (3.7) is better than the result in (3.6) for tree T.
Remark 3.3.1. From the result in [52], we can conclude that the upper bound on LE in (3.11) is better than the upper bound in (3.10). Now we will show that the upper bound on LE in (3.12) is better than the upper bound in (3.11). Thus we have to show that In this section, we give an upper bound on Laplacian energy of graph G. Now we recall some known upper bounds on LE of graphs G. In [52], Gutman et al. obtained the following upper bounds:
Remark 3.3.1. From the result in [52], we can conclude that the upper bound on LE in (3.11) is better than the upper bound in (3.10). Now we will show that the upper bound on LE in (3.12) is better than the upper bound in (3.11). Thus we have to show that In this section, we give an upper bound on Laplacian energy of graph G. Now we recall some known upper bounds on LE of graphs G. In [52], Gutman et al. obtained the following upper bounds:
Our result in Theorem 3.8.8 is always better than the above result for bipartite graphs. 3.4 Nordhaus-Gaddum-type results
Our result in Theorem 3.8.8 is always better than the above result for bipartite graphs. 3.4 Nordhaus-Gaddum-type results
For m <n, again by Lemma 4.2.2, we have By Theorem 4.3.1, we get the following two results.
For m <n, again by Lemma 4.2.2, we have By Theorem 4.3.1, we get the following two results.
Abreu et al. [1] proved the following upper bound on €(L@):
Abreu et al. [1] proved the following upper bound on €(L@):
threshold graphs in terms of n, m and T’, and characterization of extremal graphs. From the above two results, we get
threshold graphs in terms of n, m and T’, and characterization of extremal graphs. From the above two results, we get
Table 1. Different values of (T, m) and the corresponding unique threshold graph G.
Table 1. Different values of (T, m) and the corresponding unique threshold graph G.
Fig. 5. Two threshold graphs $? and 93 (Black vertices are dominating vertices). Chapter 5. Laplacian energy of threshold graphs
Fig. 5. Two threshold graphs $? and 93 (Black vertices are dominating vertices). Chapter 5. Laplacian energy of threshold graphs
2 Table 2. The list of the first 20 minimal Laplacian energies, here M = ">" and n > 20 Corollary 5.2.4. Let G be a connected threshold graph of order n. Ther
2 Table 2. The list of the first 20 minimal Laplacian energies, here M = ">" and n > 20 Corollary 5.2.4. Let G be a connected threshold graph of order n. Ther
From (5.15), one can see easily that Chapter 5. Laplacian energy of threshold graphs
From (5.15), one can see easily that Chapter 5. Laplacian energy of threshold graphs
Lemma 5.3.6. Let G be a connected threshold graph of order n with m edges. The: Chapter 5. Laplacian energy of threshold graphs
Lemma 5.3.6. Let G be a connected threshold graph of order n with m edges. The: Chapter 5. Laplacian energy of threshold graphs
Case (ti): k > T. Again using Lemmas 1.2.35 and 1.2.36, we can write Lemma 5.3.7. Let G be a connected threshold graph of order n with trace T and
Case (ti): k > T. Again using Lemmas 1.2.35 and 1.2.36, we can write Lemma 5.3.7. Let G be a connected threshold graph of order n with trace T and
5.3. Maximal Laplacian energy among connected threshold graphs 9:
5.3. Maximal Laplacian energy among connected threshold graphs 9:
3. Maximal Laplacian energy among connected threshold graphs 9 From the equality in (5.32), we get
3. Maximal Laplacian energy among connected threshold graphs 9 From the equality in (5.32), we get
graphs for which (6.1) holds or does not hold is not an easy task and is still an open problem [32]. In this chapter, we are not going to solve this open problem. But we offer some new relations between different energies of graphs.
graphs for which (6.1) holds or does not hold is not an easy task and is still an open problem [32]. In this chapter, we are not going to solve this open problem. But we offer some new relations between different energies of graphs.
Now we present a lower bound on LE of graph G that always is better than the lower bound in (6.6). Moreover, our result is true for any graphs. Chapter 6. Relation between graph energies
Now we present a lower bound on LE of graph G that always is better than the lower bound in (6.6). Moreover, our result is true for any graphs. Chapter 6. Relation between graph energies
Chapter 6. Relation between graph energies
Chapter 6. Relation between graph energies
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Spectral Graph Theory
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